Con este proyecto de semáforo PLC queremos dar la solución a la problemática en un punto de Pachacútec ubicado entre la Av. Arquitectos / Av. 225 se pretende utilizar los conocimientos adquiridos en el curso de sistemas de control automático, su debida interpretación y aplicación en el campo de los sistemas de control industrial. Todo lo aquí expuesto y mostrado ha sido obtenido de internet y autores relacionados a formaciones eléctricas y/o a fines.
Estos dispositivos electrónicos nos ayudan a solucionar los controles de circuitos complejos para hacerlos más sencillos y lograr la automatización de los sistemas. Para que este dispositivo trabaje es necesario que reciba una señal esto puede ser por medio de sensores que actúan sobre las salidas del dispositivo está recibirá una señal de entrada y el programa podrá convertir a una señal de salida los autómatas llegaron para simplificar los circuitos eléctricos como las instalaciones eléctricas, las subestaciones y la industria en general ya que con este tipo de dispositivos se pueden controlar puertas, motores entre otros sistemas.
SISTEMA DE ALUMBRADO LED EN UN ALMACÉN LOGÍSTICO - FUNDACIÓN PACHACUTECFundación Pachacutec
El diseño e implementación de un sistema de alumbrado en un almacén logístico eficiente y sostenible para mejorar la seguridad, la eficiencia energética y la calidad de trabajo.
El estudio aborda el desafío actual en el sistema de iluminación en el almacén logístico, como el consumo de energía excesivo, el encendido manual de las mismas.
La investigación comienza con un análisis de las tecnologías de iluminación existentes, evaluando su rendimiento, sostenibilidad y costos. Se consideran tecnologías emergentes, como la iluminación LED y las soluciones inteligentes los cuales inician el funcionamiento de estas mismas, con el objetivo de proponer un sistema que optimice el funcionamiento en el sistema de iluminación.
El diseño del sistema se basa en el cambio de las luminarias actuales, las cuales están fabricadas a base de su componente principal el halogenuro de Mercurio, por luminarias Leds, las cuales nos brindan los mismos beneficios, pero con un ahorro del 50% por ciento al actual. y hacer el uso de la automatización y la capacidad de ajuste en tiempo real permiten adaptar la intensidad luminosa según las condiciones y los patrones de actividad, contribuyendo así al ahorro de energía y dinero.
CIRCUITO PLC PARA SIMULAR JUEGO CANTA Y GANA EN EL INSTITUTO FUNDACIÓN PACHAC...Fundación Pachacutec
El presente proyecto elaborado en el dispositivo PLC (controlador lógico programable) LOGO se basa en los juegos de canta y gana o shows de preguntas y respuestas en los cuales los participantes pulsan botones de diversos colores, encendiendo solo el primero en ser presionado mientras que los demás quedan en estado abierto, de ese modo podremos saber cuál participante fue más rápido en reaccionar tras el aviso del presentador.
La finalidad del proyecto es poder elaborar un circuito de manera rápida sin tanto problema y de ese modo pasar un momento de diversión con los alumnos, compañeros o profesores ya que éste trabajo se puede usar tanto para un momento de ocio o para poder poner a prueba a los alumnos mediante preguntas de conocimiento.
SISTEMAS DE CONTROL DE 2 BOMBAS ALTERNADAS POR MEDIO DE UN DISPOSITIVO MOVIL ...Fundación Pachacutec
El desarrollo del proyecto se enfocó en la creación de un sistema de control automatizado para dos bombas alternadas, controladas inalámbricamente por medio de un dispositivo móvil. El objetivo general del proyecto fue diseñar, implementar y evaluar un sistema de alternancia de dos bombas controladas por un PLC, cuyo monitoreo y control sean accesibles mediante una aplicación móvil. Los objetivos específicos incluyeron desarrollar un sistema de control que garantice la operación segura y eficiente de dos bombas en paralelo e implementar una aplicación móvil que permita a los operadores controlar el sistema en tiempo real. El proyecto se basó en la teoría de procesos, que se define como el conjunto de actividades y recursos interrelacionados que transforman los elementos de entrada en elementos de salida aportando valor para el usuario. Se utilizó un PLC para controlar las bombas y se diseñó un esquema de mando y fuerza para el sistema. Además, se programó el sistema para permitir el control manual y automático de las bombas. En cuanto a la conexión inalámbrica, se detalló la configuración del Router WIFI y la configuración de red necesarias para el correcto funcionamiento del sistema. También se describió la conexión entre el Router y el PLC.
2024 State of Marketing Report – by HubspotMarius Sescu
https://www.hubspot.com/state-of-marketing
· Scaling relationships and proving ROI
· Social media is the place for search, sales, and service
· Authentic influencer partnerships fuel brand growth
· The strongest connections happen via call, click, chat, and camera.
· Time saved with AI leads to more creative work
· Seeking: A single source of truth
· TLDR; Get on social, try AI, and align your systems.
· More human marketing, powered by robots
ChatGPT is a revolutionary addition to the world since its introduction in 2022. A big shift in the sector of information gathering and processing happened because of this chatbot. What is the story of ChatGPT? How is the bot responding to prompts and generating contents? Swipe through these slides prepared by Expeed Software, a web development company regarding the development and technical intricacies of ChatGPT!
Product Design Trends in 2024 | Teenage EngineeringsPixeldarts
The realm of product design is a constantly changing environment where technology and style intersect. Every year introduces fresh challenges and exciting trends that mold the future of this captivating art form. In this piece, we delve into the significant trends set to influence the look and functionality of product design in the year 2024.
How Race, Age and Gender Shape Attitudes Towards Mental HealthThinkNow
Mental health has been in the news quite a bit lately. Dozens of U.S. states are currently suing Meta for contributing to the youth mental health crisis by inserting addictive features into their products, while the U.S. Surgeon General is touring the nation to bring awareness to the growing epidemic of loneliness and isolation. The country has endured periods of low national morale, such as in the 1970s when high inflation and the energy crisis worsened public sentiment following the Vietnam War. The current mood, however, feels different. Gallup recently reported that national mental health is at an all-time low, with few bright spots to lift spirits.
To better understand how Americans are feeling and their attitudes towards mental health in general, ThinkNow conducted a nationally representative quantitative survey of 1,500 respondents and found some interesting differences among ethnic, age and gender groups.
Technology
For example, 52% agree that technology and social media have a negative impact on mental health, but when broken out by race, 61% of Whites felt technology had a negative effect, and only 48% of Hispanics thought it did.
While technology has helped us keep in touch with friends and family in faraway places, it appears to have degraded our ability to connect in person. Staying connected online is a double-edged sword since the same news feed that brings us pictures of the grandkids and fluffy kittens also feeds us news about the wars in Israel and Ukraine, the dysfunction in Washington, the latest mass shooting and the climate crisis.
Hispanics may have a built-in defense against the isolation technology breeds, owing to their large, multigenerational households, strong social support systems, and tendency to use social media to stay connected with relatives abroad.
Age and Gender
When asked how individuals rate their mental health, men rate it higher than women by 11 percentage points, and Baby Boomers rank it highest at 83%, saying it’s good or excellent vs. 57% of Gen Z saying the same.
Gen Z spends the most amount of time on social media, so the notion that social media negatively affects mental health appears to be correlated. Unfortunately, Gen Z is also the generation that’s least comfortable discussing mental health concerns with healthcare professionals. Only 40% of them state they’re comfortable discussing their issues with a professional compared to 60% of Millennials and 65% of Boomers.
Race Affects Attitudes
As seen in previous research conducted by ThinkNow, Asian Americans lag other groups when it comes to awareness of mental health issues. Twenty-four percent of Asian Americans believe that having a mental health issue is a sign of weakness compared to the 16% average for all groups. Asians are also considerably less likely to be aware of mental health services in their communities (42% vs. 55%) and most likely to seek out information on social media (51% vs. 35%).
Con este proyecto de semáforo PLC queremos dar la solución a la problemática en un punto de Pachacútec ubicado entre la Av. Arquitectos / Av. 225 se pretende utilizar los conocimientos adquiridos en el curso de sistemas de control automático, su debida interpretación y aplicación en el campo de los sistemas de control industrial. Todo lo aquí expuesto y mostrado ha sido obtenido de internet y autores relacionados a formaciones eléctricas y/o a fines.
Estos dispositivos electrónicos nos ayudan a solucionar los controles de circuitos complejos para hacerlos más sencillos y lograr la automatización de los sistemas. Para que este dispositivo trabaje es necesario que reciba una señal esto puede ser por medio de sensores que actúan sobre las salidas del dispositivo está recibirá una señal de entrada y el programa podrá convertir a una señal de salida los autómatas llegaron para simplificar los circuitos eléctricos como las instalaciones eléctricas, las subestaciones y la industria en general ya que con este tipo de dispositivos se pueden controlar puertas, motores entre otros sistemas.
SISTEMA DE ALUMBRADO LED EN UN ALMACÉN LOGÍSTICO - FUNDACIÓN PACHACUTECFundación Pachacutec
El diseño e implementación de un sistema de alumbrado en un almacén logístico eficiente y sostenible para mejorar la seguridad, la eficiencia energética y la calidad de trabajo.
El estudio aborda el desafío actual en el sistema de iluminación en el almacén logístico, como el consumo de energía excesivo, el encendido manual de las mismas.
La investigación comienza con un análisis de las tecnologías de iluminación existentes, evaluando su rendimiento, sostenibilidad y costos. Se consideran tecnologías emergentes, como la iluminación LED y las soluciones inteligentes los cuales inician el funcionamiento de estas mismas, con el objetivo de proponer un sistema que optimice el funcionamiento en el sistema de iluminación.
El diseño del sistema se basa en el cambio de las luminarias actuales, las cuales están fabricadas a base de su componente principal el halogenuro de Mercurio, por luminarias Leds, las cuales nos brindan los mismos beneficios, pero con un ahorro del 50% por ciento al actual. y hacer el uso de la automatización y la capacidad de ajuste en tiempo real permiten adaptar la intensidad luminosa según las condiciones y los patrones de actividad, contribuyendo así al ahorro de energía y dinero.
CIRCUITO PLC PARA SIMULAR JUEGO CANTA Y GANA EN EL INSTITUTO FUNDACIÓN PACHAC...Fundación Pachacutec
El presente proyecto elaborado en el dispositivo PLC (controlador lógico programable) LOGO se basa en los juegos de canta y gana o shows de preguntas y respuestas en los cuales los participantes pulsan botones de diversos colores, encendiendo solo el primero en ser presionado mientras que los demás quedan en estado abierto, de ese modo podremos saber cuál participante fue más rápido en reaccionar tras el aviso del presentador.
La finalidad del proyecto es poder elaborar un circuito de manera rápida sin tanto problema y de ese modo pasar un momento de diversión con los alumnos, compañeros o profesores ya que éste trabajo se puede usar tanto para un momento de ocio o para poder poner a prueba a los alumnos mediante preguntas de conocimiento.
SISTEMAS DE CONTROL DE 2 BOMBAS ALTERNADAS POR MEDIO DE UN DISPOSITIVO MOVIL ...Fundación Pachacutec
El desarrollo del proyecto se enfocó en la creación de un sistema de control automatizado para dos bombas alternadas, controladas inalámbricamente por medio de un dispositivo móvil. El objetivo general del proyecto fue diseñar, implementar y evaluar un sistema de alternancia de dos bombas controladas por un PLC, cuyo monitoreo y control sean accesibles mediante una aplicación móvil. Los objetivos específicos incluyeron desarrollar un sistema de control que garantice la operación segura y eficiente de dos bombas en paralelo e implementar una aplicación móvil que permita a los operadores controlar el sistema en tiempo real. El proyecto se basó en la teoría de procesos, que se define como el conjunto de actividades y recursos interrelacionados que transforman los elementos de entrada en elementos de salida aportando valor para el usuario. Se utilizó un PLC para controlar las bombas y se diseñó un esquema de mando y fuerza para el sistema. Además, se programó el sistema para permitir el control manual y automático de las bombas. En cuanto a la conexión inalámbrica, se detalló la configuración del Router WIFI y la configuración de red necesarias para el correcto funcionamiento del sistema. También se describió la conexión entre el Router y el PLC.
2024 State of Marketing Report – by HubspotMarius Sescu
https://www.hubspot.com/state-of-marketing
· Scaling relationships and proving ROI
· Social media is the place for search, sales, and service
· Authentic influencer partnerships fuel brand growth
· The strongest connections happen via call, click, chat, and camera.
· Time saved with AI leads to more creative work
· Seeking: A single source of truth
· TLDR; Get on social, try AI, and align your systems.
· More human marketing, powered by robots
ChatGPT is a revolutionary addition to the world since its introduction in 2022. A big shift in the sector of information gathering and processing happened because of this chatbot. What is the story of ChatGPT? How is the bot responding to prompts and generating contents? Swipe through these slides prepared by Expeed Software, a web development company regarding the development and technical intricacies of ChatGPT!
Product Design Trends in 2024 | Teenage EngineeringsPixeldarts
The realm of product design is a constantly changing environment where technology and style intersect. Every year introduces fresh challenges and exciting trends that mold the future of this captivating art form. In this piece, we delve into the significant trends set to influence the look and functionality of product design in the year 2024.
How Race, Age and Gender Shape Attitudes Towards Mental HealthThinkNow
Mental health has been in the news quite a bit lately. Dozens of U.S. states are currently suing Meta for contributing to the youth mental health crisis by inserting addictive features into their products, while the U.S. Surgeon General is touring the nation to bring awareness to the growing epidemic of loneliness and isolation. The country has endured periods of low national morale, such as in the 1970s when high inflation and the energy crisis worsened public sentiment following the Vietnam War. The current mood, however, feels different. Gallup recently reported that national mental health is at an all-time low, with few bright spots to lift spirits.
To better understand how Americans are feeling and their attitudes towards mental health in general, ThinkNow conducted a nationally representative quantitative survey of 1,500 respondents and found some interesting differences among ethnic, age and gender groups.
Technology
For example, 52% agree that technology and social media have a negative impact on mental health, but when broken out by race, 61% of Whites felt technology had a negative effect, and only 48% of Hispanics thought it did.
While technology has helped us keep in touch with friends and family in faraway places, it appears to have degraded our ability to connect in person. Staying connected online is a double-edged sword since the same news feed that brings us pictures of the grandkids and fluffy kittens also feeds us news about the wars in Israel and Ukraine, the dysfunction in Washington, the latest mass shooting and the climate crisis.
Hispanics may have a built-in defense against the isolation technology breeds, owing to their large, multigenerational households, strong social support systems, and tendency to use social media to stay connected with relatives abroad.
Age and Gender
When asked how individuals rate their mental health, men rate it higher than women by 11 percentage points, and Baby Boomers rank it highest at 83%, saying it’s good or excellent vs. 57% of Gen Z saying the same.
Gen Z spends the most amount of time on social media, so the notion that social media negatively affects mental health appears to be correlated. Unfortunately, Gen Z is also the generation that’s least comfortable discussing mental health concerns with healthcare professionals. Only 40% of them state they’re comfortable discussing their issues with a professional compared to 60% of Millennials and 65% of Boomers.
Race Affects Attitudes
As seen in previous research conducted by ThinkNow, Asian Americans lag other groups when it comes to awareness of mental health issues. Twenty-four percent of Asian Americans believe that having a mental health issue is a sign of weakness compared to the 16% average for all groups. Asians are also considerably less likely to be aware of mental health services in their communities (42% vs. 55%) and most likely to seek out information on social media (51% vs. 35%).
AI Trends in Creative Operations 2024 by Artwork Flow.pdfmarketingartwork
Creative operations teams expect increased AI use in 2024. Currently, over half of tasks are not AI-enabled, but this is expected to decrease in the coming year. ChatGPT is the most popular AI tool currently. Business leaders are more actively exploring AI benefits than individual contributors. Most respondents do not believe AI will impact workforce size in 2024. However, some inhibitions still exist around AI accuracy and lack of understanding. Creatives primarily want to use AI to save time on mundane tasks and boost productivity.
Organizational culture includes values, norms, systems, symbols, language, assumptions, beliefs, and habits that influence employee behaviors and how people interpret those behaviors. It is important because culture can help or hinder a company's success. Some key aspects of Netflix's culture that help it achieve results include hiring smartly so every position has stars, focusing on attitude over just aptitude, and having a strict policy against peacocks, whiners, and jerks.
PEPSICO Presentation to CAGNY Conference Feb 2024Neil Kimberley
PepsiCo provided a safe harbor statement noting that any forward-looking statements are based on currently available information and are subject to risks and uncertainties. It also provided information on non-GAAP measures and directing readers to its website for disclosure and reconciliation. The document then discussed PepsiCo's business overview, including that it is a global beverage and convenient food company with iconic brands, $91 billion in net revenue in 2023, and nearly $14 billion in core operating profit. It operates through a divisional structure with a focus on local consumers.
Content Methodology: A Best Practices Report (Webinar)contently
This document provides an overview of content methodology best practices. It defines content methodology as establishing objectives, KPIs, and a culture of continuous learning and iteration. An effective methodology focuses on connecting with audiences, creating optimal content, and optimizing processes. It also discusses why a methodology is needed due to the competitive landscape, proliferation of channels, and opportunities for improvement. Components of an effective methodology include defining objectives and KPIs, audience analysis, identifying opportunities, and evaluating resources. The document concludes with recommendations around creating a content plan, testing and optimizing content over 90 days.
How to Prepare For a Successful Job Search for 2024Albert Qian
The document provides guidance on preparing a job search for 2024. It discusses the state of the job market, focusing on growth in AI and healthcare but also continued layoffs. It recommends figuring out what you want to do by researching interests and skills, then conducting informational interviews. The job search should involve building a personal brand on LinkedIn, actively applying to jobs, tailoring resumes and interviews, maintaining job hunting as a habit, and continuing self-improvement. Once hired, the document advises setting new goals and keeping skills and networking active in case of future opportunities.
A report by thenetworkone and Kurio.
The contributing experts and agencies are (in an alphabetical order): Sylwia Rytel, Social Media Supervisor, 180heartbeats + JUNG v MATT (PL), Sharlene Jenner, Vice President - Director of Engagement Strategy, Abelson Taylor (USA), Alex Casanovas, Digital Director, Atrevia (ES), Dora Beilin, Senior Social Strategist, Barrett Hoffher (USA), Min Seo, Campaign Director, Brand New Agency (KR), Deshé M. Gully, Associate Strategist, Day One Agency (USA), Francesca Trevisan, Strategist, Different (IT), Trevor Crossman, CX and Digital Transformation Director; Olivia Hussey, Strategic Planner; Simi Srinarula, Social Media Manager, The Hallway (AUS), James Hebbert, Managing Director, Hylink (CN / UK), Mundy Álvarez, Planning Director; Pedro Rojas, Social Media Manager; Pancho González, CCO, Inbrax (CH), Oana Oprea, Head of Digital Planning, Jam Session Agency (RO), Amy Bottrill, Social Account Director, Launch (UK), Gaby Arriaga, Founder, Leonardo1452 (MX), Shantesh S Row, Creative Director, Liwa (UAE), Rajesh Mehta, Chief Strategy Officer; Dhruv Gaur, Digital Planning Lead; Leonie Mergulhao, Account Supervisor - Social Media & PR, Medulla (IN), Aurelija Plioplytė, Head of Digital & Social, Not Perfect (LI), Daiana Khaidargaliyeva, Account Manager, Osaka Labs (UK / USA), Stefanie Söhnchen, Vice President Digital, PIABO Communications (DE), Elisabeth Winiartati, Managing Consultant, Head of Global Integrated Communications; Lydia Aprina, Account Manager, Integrated Marketing and Communications; Nita Prabowo, Account Manager, Integrated Marketing and Communications; Okhi, Web Developer, PNTR Group (ID), Kei Obusan, Insights Director; Daffi Ranandi, Insights Manager, Radarr (SG), Gautam Reghunath, Co-founder & CEO, Talented (IN), Donagh Humphreys, Head of Social and Digital Innovation, THINKHOUSE (IRE), Sarah Yim, Strategy Director, Zulu Alpha Kilo (CA).
Trends In Paid Search: Navigating The Digital Landscape In 2024Search Engine Journal
The search marketing landscape is evolving rapidly with new technologies, and professionals, like you, rely on innovative paid search strategies to meet changing demands.
It’s important that you’re ready to implement new strategies in 2024.
Check this out and learn the top trends in paid search advertising that are expected to gain traction, so you can drive higher ROI more efficiently in 2024.
You’ll learn:
- The latest trends in AI and automation, and what this means for an evolving paid search ecosystem.
- New developments in privacy and data regulation.
- Emerging ad formats that are expected to make an impact next year.
Watch Sreekant Lanka from iQuanti and Irina Klein from OneMain Financial as they dive into the future of paid search and explore the trends, strategies, and technologies that will shape the search marketing landscape.
If you’re looking to assess your paid search strategy and design an industry-aligned plan for 2024, then this webinar is for you.
5 Public speaking tips from TED - Visualized summarySpeakerHub
From their humble beginnings in 1984, TED has grown into the world’s most powerful amplifier for speakers and thought-leaders to share their ideas. They have over 2,400 filmed talks (not including the 30,000+ TEDx videos) freely available online, and have hosted over 17,500 events around the world.
With over one billion views in a year, it’s no wonder that so many speakers are looking to TED for ideas on how to share their message more effectively.
The article “5 Public-Speaking Tips TED Gives Its Speakers”, by Carmine Gallo for Forbes, gives speakers five practical ways to connect with their audience, and effectively share their ideas on stage.
Whether you are gearing up to get on a TED stage yourself, or just want to master the skills that so many of their speakers possess, these tips and quotes from Chris Anderson, the TED Talks Curator, will encourage you to make the most impactful impression on your audience.
See the full article and more summaries like this on SpeakerHub here: https://speakerhub.com/blog/5-presentation-tips-ted-gives-its-speakers
See the original article on Forbes here:
http://www.forbes.com/forbes/welcome/?toURL=http://www.forbes.com/sites/carminegallo/2016/05/06/5-public-speaking-tips-ted-gives-its-speakers/&refURL=&referrer=#5c07a8221d9b
ChatGPT and the Future of Work - Clark Boyd Clark Boyd
Everyone is in agreement that ChatGPT (and other generative AI tools) will shape the future of work. Yet there is little consensus on exactly how, when, and to what extent this technology will change our world.
Businesses that extract maximum value from ChatGPT will use it as a collaborative tool for everything from brainstorming to technical maintenance.
For individuals, now is the time to pinpoint the skills the future professional will need to thrive in the AI age.
Check out this presentation to understand what ChatGPT is, how it will shape the future of work, and how you can prepare to take advantage.
The document provides career advice for getting into the tech field, including:
- Doing projects and internships in college to build a portfolio.
- Learning about different roles and technologies through industry research.
- Contributing to open source projects to build experience and network.
- Developing a personal brand through a website and social media presence.
- Networking through events, communities, and finding a mentor.
- Practicing interviews through mock interviews and whiteboarding coding questions.
Google's Just Not That Into You: Understanding Core Updates & Search IntentLily Ray
1. Core updates from Google periodically change how its algorithms assess and rank websites and pages. This can impact rankings through shifts in user intent, site quality issues being caught up to, world events influencing queries, and overhauls to search like the E-A-T framework.
2. There are many possible user intents beyond just transactional, navigational and informational. Identifying intent shifts is important during core updates. Sites may need to optimize for new intents through different content types and sections.
3. Responding effectively to core updates requires analyzing "before and after" data to understand changes, identifying new intents or page types, and ensuring content matches appropriate intents across video, images, knowledge graphs and more.
A brief introduction to DataScience with explaining of the concepts, algorithms, machine learning, supervised and unsupervised learning, clustering, statistics, data preprocessing, real-world applications etc.
It's part of a Data Science Corner Campaign where I will be discussing the fundamentals of DataScience, AIML, Statistics etc.
Time Management & Productivity - Best PracticesVit Horky
Here's my presentation on by proven best practices how to manage your work time effectively and how to improve your productivity. It includes practical tips and how to use tools such as Slack, Google Apps, Hubspot, Google Calendar, Gmail and others.
The six step guide to practical project managementMindGenius
The six step guide to practical project management
If you think managing projects is too difficult, think again.
We’ve stripped back project management processes to the
basics – to make it quicker and easier, without sacrificing
the vital ingredients for success.
“If you’re looking for some real-world guidance, then The Six Step Guide to Practical Project Management will help.”
Dr Andrew Makar, Tactical Project Management
Unlocking the Power of ChatGPT and AI in Testing - A Real-World Look, present...Applitools
During this webinar, Anand Bagmar demonstrates how AI tools such as ChatGPT can be applied to various stages of the software development life cycle (SDLC) using an eCommerce application case study. Find the on-demand recording and more info at https://applitools.info/b59
Key takeaways:
• Learn how to use ChatGPT to add AI power to your testing and test automation
• Understand the limitations of the technology and where human expertise is crucial
• Gain insight into different AI-based tools
• Adopt AI-based tools to stay relevant and optimize work for developers and testers
* ChatGPT and OpenAI belong to OpenAI, L.L.C.
The document discusses various AI tools from OpenAI like GPT-3 and DALL-E 2, as well as ChatGPT. It explores how search engines are using AI and things to consider around AI-generated content. Potential SEO uses of ChatGPT are also presented, such as generating content at scale, conducting topic research, and automating basic coding tasks. The document encourages further reading on using ChatGPT for SEO purposes.
More than Just Lines on a Map: Best Practices for U.S Bike Routes
This session highlights best practices and lessons learned for U.S. Bike Route System designation, as well as how and why these routes should be integrated into bicycle planning at the local and regional level.
Presenters:
Presenter: Kevin Luecke Toole Design Group
Co-Presenter: Virginia Sullivan Adventure Cycling Association
AI Trends in Creative Operations 2024 by Artwork Flow.pdfmarketingartwork
Creative operations teams expect increased AI use in 2024. Currently, over half of tasks are not AI-enabled, but this is expected to decrease in the coming year. ChatGPT is the most popular AI tool currently. Business leaders are more actively exploring AI benefits than individual contributors. Most respondents do not believe AI will impact workforce size in 2024. However, some inhibitions still exist around AI accuracy and lack of understanding. Creatives primarily want to use AI to save time on mundane tasks and boost productivity.
Organizational culture includes values, norms, systems, symbols, language, assumptions, beliefs, and habits that influence employee behaviors and how people interpret those behaviors. It is important because culture can help or hinder a company's success. Some key aspects of Netflix's culture that help it achieve results include hiring smartly so every position has stars, focusing on attitude over just aptitude, and having a strict policy against peacocks, whiners, and jerks.
PEPSICO Presentation to CAGNY Conference Feb 2024Neil Kimberley
PepsiCo provided a safe harbor statement noting that any forward-looking statements are based on currently available information and are subject to risks and uncertainties. It also provided information on non-GAAP measures and directing readers to its website for disclosure and reconciliation. The document then discussed PepsiCo's business overview, including that it is a global beverage and convenient food company with iconic brands, $91 billion in net revenue in 2023, and nearly $14 billion in core operating profit. It operates through a divisional structure with a focus on local consumers.
Content Methodology: A Best Practices Report (Webinar)contently
This document provides an overview of content methodology best practices. It defines content methodology as establishing objectives, KPIs, and a culture of continuous learning and iteration. An effective methodology focuses on connecting with audiences, creating optimal content, and optimizing processes. It also discusses why a methodology is needed due to the competitive landscape, proliferation of channels, and opportunities for improvement. Components of an effective methodology include defining objectives and KPIs, audience analysis, identifying opportunities, and evaluating resources. The document concludes with recommendations around creating a content plan, testing and optimizing content over 90 days.
How to Prepare For a Successful Job Search for 2024Albert Qian
The document provides guidance on preparing a job search for 2024. It discusses the state of the job market, focusing on growth in AI and healthcare but also continued layoffs. It recommends figuring out what you want to do by researching interests and skills, then conducting informational interviews. The job search should involve building a personal brand on LinkedIn, actively applying to jobs, tailoring resumes and interviews, maintaining job hunting as a habit, and continuing self-improvement. Once hired, the document advises setting new goals and keeping skills and networking active in case of future opportunities.
A report by thenetworkone and Kurio.
The contributing experts and agencies are (in an alphabetical order): Sylwia Rytel, Social Media Supervisor, 180heartbeats + JUNG v MATT (PL), Sharlene Jenner, Vice President - Director of Engagement Strategy, Abelson Taylor (USA), Alex Casanovas, Digital Director, Atrevia (ES), Dora Beilin, Senior Social Strategist, Barrett Hoffher (USA), Min Seo, Campaign Director, Brand New Agency (KR), Deshé M. Gully, Associate Strategist, Day One Agency (USA), Francesca Trevisan, Strategist, Different (IT), Trevor Crossman, CX and Digital Transformation Director; Olivia Hussey, Strategic Planner; Simi Srinarula, Social Media Manager, The Hallway (AUS), James Hebbert, Managing Director, Hylink (CN / UK), Mundy Álvarez, Planning Director; Pedro Rojas, Social Media Manager; Pancho González, CCO, Inbrax (CH), Oana Oprea, Head of Digital Planning, Jam Session Agency (RO), Amy Bottrill, Social Account Director, Launch (UK), Gaby Arriaga, Founder, Leonardo1452 (MX), Shantesh S Row, Creative Director, Liwa (UAE), Rajesh Mehta, Chief Strategy Officer; Dhruv Gaur, Digital Planning Lead; Leonie Mergulhao, Account Supervisor - Social Media & PR, Medulla (IN), Aurelija Plioplytė, Head of Digital & Social, Not Perfect (LI), Daiana Khaidargaliyeva, Account Manager, Osaka Labs (UK / USA), Stefanie Söhnchen, Vice President Digital, PIABO Communications (DE), Elisabeth Winiartati, Managing Consultant, Head of Global Integrated Communications; Lydia Aprina, Account Manager, Integrated Marketing and Communications; Nita Prabowo, Account Manager, Integrated Marketing and Communications; Okhi, Web Developer, PNTR Group (ID), Kei Obusan, Insights Director; Daffi Ranandi, Insights Manager, Radarr (SG), Gautam Reghunath, Co-founder & CEO, Talented (IN), Donagh Humphreys, Head of Social and Digital Innovation, THINKHOUSE (IRE), Sarah Yim, Strategy Director, Zulu Alpha Kilo (CA).
Trends In Paid Search: Navigating The Digital Landscape In 2024Search Engine Journal
The search marketing landscape is evolving rapidly with new technologies, and professionals, like you, rely on innovative paid search strategies to meet changing demands.
It’s important that you’re ready to implement new strategies in 2024.
Check this out and learn the top trends in paid search advertising that are expected to gain traction, so you can drive higher ROI more efficiently in 2024.
You’ll learn:
- The latest trends in AI and automation, and what this means for an evolving paid search ecosystem.
- New developments in privacy and data regulation.
- Emerging ad formats that are expected to make an impact next year.
Watch Sreekant Lanka from iQuanti and Irina Klein from OneMain Financial as they dive into the future of paid search and explore the trends, strategies, and technologies that will shape the search marketing landscape.
If you’re looking to assess your paid search strategy and design an industry-aligned plan for 2024, then this webinar is for you.
5 Public speaking tips from TED - Visualized summarySpeakerHub
From their humble beginnings in 1984, TED has grown into the world’s most powerful amplifier for speakers and thought-leaders to share their ideas. They have over 2,400 filmed talks (not including the 30,000+ TEDx videos) freely available online, and have hosted over 17,500 events around the world.
With over one billion views in a year, it’s no wonder that so many speakers are looking to TED for ideas on how to share their message more effectively.
The article “5 Public-Speaking Tips TED Gives Its Speakers”, by Carmine Gallo for Forbes, gives speakers five practical ways to connect with their audience, and effectively share their ideas on stage.
Whether you are gearing up to get on a TED stage yourself, or just want to master the skills that so many of their speakers possess, these tips and quotes from Chris Anderson, the TED Talks Curator, will encourage you to make the most impactful impression on your audience.
See the full article and more summaries like this on SpeakerHub here: https://speakerhub.com/blog/5-presentation-tips-ted-gives-its-speakers
See the original article on Forbes here:
http://www.forbes.com/forbes/welcome/?toURL=http://www.forbes.com/sites/carminegallo/2016/05/06/5-public-speaking-tips-ted-gives-its-speakers/&refURL=&referrer=#5c07a8221d9b
ChatGPT and the Future of Work - Clark Boyd Clark Boyd
Everyone is in agreement that ChatGPT (and other generative AI tools) will shape the future of work. Yet there is little consensus on exactly how, when, and to what extent this technology will change our world.
Businesses that extract maximum value from ChatGPT will use it as a collaborative tool for everything from brainstorming to technical maintenance.
For individuals, now is the time to pinpoint the skills the future professional will need to thrive in the AI age.
Check out this presentation to understand what ChatGPT is, how it will shape the future of work, and how you can prepare to take advantage.
The document provides career advice for getting into the tech field, including:
- Doing projects and internships in college to build a portfolio.
- Learning about different roles and technologies through industry research.
- Contributing to open source projects to build experience and network.
- Developing a personal brand through a website and social media presence.
- Networking through events, communities, and finding a mentor.
- Practicing interviews through mock interviews and whiteboarding coding questions.
Google's Just Not That Into You: Understanding Core Updates & Search IntentLily Ray
1. Core updates from Google periodically change how its algorithms assess and rank websites and pages. This can impact rankings through shifts in user intent, site quality issues being caught up to, world events influencing queries, and overhauls to search like the E-A-T framework.
2. There are many possible user intents beyond just transactional, navigational and informational. Identifying intent shifts is important during core updates. Sites may need to optimize for new intents through different content types and sections.
3. Responding effectively to core updates requires analyzing "before and after" data to understand changes, identifying new intents or page types, and ensuring content matches appropriate intents across video, images, knowledge graphs and more.
A brief introduction to DataScience with explaining of the concepts, algorithms, machine learning, supervised and unsupervised learning, clustering, statistics, data preprocessing, real-world applications etc.
It's part of a Data Science Corner Campaign where I will be discussing the fundamentals of DataScience, AIML, Statistics etc.
Time Management & Productivity - Best PracticesVit Horky
Here's my presentation on by proven best practices how to manage your work time effectively and how to improve your productivity. It includes practical tips and how to use tools such as Slack, Google Apps, Hubspot, Google Calendar, Gmail and others.
The six step guide to practical project managementMindGenius
The six step guide to practical project management
If you think managing projects is too difficult, think again.
We’ve stripped back project management processes to the
basics – to make it quicker and easier, without sacrificing
the vital ingredients for success.
“If you’re looking for some real-world guidance, then The Six Step Guide to Practical Project Management will help.”
Dr Andrew Makar, Tactical Project Management
Unlocking the Power of ChatGPT and AI in Testing - A Real-World Look, present...Applitools
During this webinar, Anand Bagmar demonstrates how AI tools such as ChatGPT can be applied to various stages of the software development life cycle (SDLC) using an eCommerce application case study. Find the on-demand recording and more info at https://applitools.info/b59
Key takeaways:
• Learn how to use ChatGPT to add AI power to your testing and test automation
• Understand the limitations of the technology and where human expertise is crucial
• Gain insight into different AI-based tools
• Adopt AI-based tools to stay relevant and optimize work for developers and testers
* ChatGPT and OpenAI belong to OpenAI, L.L.C.
The document discusses various AI tools from OpenAI like GPT-3 and DALL-E 2, as well as ChatGPT. It explores how search engines are using AI and things to consider around AI-generated content. Potential SEO uses of ChatGPT are also presented, such as generating content at scale, conducting topic research, and automating basic coding tasks. The document encourages further reading on using ChatGPT for SEO purposes.
More than Just Lines on a Map: Best Practices for U.S Bike Routes
This session highlights best practices and lessons learned for U.S. Bike Route System designation, as well as how and why these routes should be integrated into bicycle planning at the local and regional level.
Presenters:
Presenter: Kevin Luecke Toole Design Group
Co-Presenter: Virginia Sullivan Adventure Cycling Association
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23. CHAPTER 2. ALGEBRA
13
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81. ( ) +
+
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25. CHAPTER 2. ALGEBRA
15
2.4 Roots
=
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~
~Ä = =
=
92. åã å
ã
ã
å
Ä
~
Ä
~ = =
=
93. å
å
å
Ä
~
Ä
~
= I= M
Ä ≠ =
=
94. åã
å
ã
åã å
åã ã
ã
å
Ä
~
Ä
~
Ä
~
=
= I= M
Ä ≠ K=
=
95. ( ) å ãé
é
å ã
~
~ = =
=
96. ( ) ~
~
å
å
= =
=
97.
åé ãé
å ã
~
~ = =
=
98. å
ã
å ã
~
~ = =
=
99. ãå
ã å
~
~ = =
=
100. ( ) å ã
ã
å
~
~ = =
=
26. CHAPTER 2. ALGEBRA
16
101.
~
~
~
N å N
å
å
−
= I= M
~ ≠ K=
=
102.
O
Ä
~
~
O
Ä
~
~
Ä
~
O
O
−
−
±
−
+
=
± =
=
103.
Ä
~
Ä
~
Ä
~
N
−
=
±
m
=
=
=
=
2.5 Logarithms
=
mçëáíáîÉ=êÉ~ä=åìãÄÉêëW=ñI=óI=~I=ÅI=â=
k~íìê~ä=åìãÄÉêW=å==
=
=
104. aÉÑáåáíáçå=çÑ=içÖ~êáíÜã=
ñ
äçÖ
ó ~
= =áÑ=~åÇ=çåäó=áÑ= ó
~
ñ = I= M
~ > I= N
~ ≠ K=
=
105. M
N
äçÖ~ = =
=
106. N
~
äçÖ~ = =
=
107.
<
∞
+
>
∞
−
=
N
~
áÑ
N
~
áÑ
M
äçÖ~ =
=
108. ( ) ó
äçÖ
ñ
äçÖ
ñó
äçÖ ~
~
~ +
= =
=
109. ó
äçÖ
ñ
äçÖ
ó
ñ
äçÖ ~
~
~ −
= =
27. CHAPTER 2. ALGEBRA
17
110. ( ) ñ
äçÖ
å
ñ
äçÖ ~
å
~ = =
=
111. ñ
äçÖ
å
N
ñ
äçÖ ~
å
~ = =
=
112. Å
äçÖ
ñ
äçÖ
~
äçÖ
ñ
äçÖ
ñ
äçÖ ~
Å
Å
Å
~ ⋅
=
= I= M
Å > I= N
Å ≠ K=
=
113.
~
äçÖ
N
Å
äçÖ
Å
~ = =
=
114. ñ
äçÖ~
~
ñ = =
=
115. içÖ~êáíÜã=íç=_~ëÉ=NM=
ñ
äçÖ
ñ
äçÖNM = =
=
116. k~íìê~ä=içÖ~êáíÜã=
ñ
äå
ñ
äçÖÉ = I==
ïÜÉêÉ= K
TNUOUNUOU
K
O
â
N
N
äáã
É
â
â
=
+
=
∞
→
=
=
117. ñ
äå
QPQOVQ
K
M
ñ
äå
NM
äå
N
ñ
äçÖ =
= =
=
118. ñ
äçÖ
PMORUR
K
O
ñ
äçÖ
É
äçÖ
N
ñ
äå =
= =
=
=
=
=
=
28. CHAPTER 2. ALGEBRA
18
2.6 Equations
=
oÉ~ä=åìãÄÉêëW=~I=ÄI=ÅI=éI=èI=ìI=î=
pçäìíáçåëW= N
ñ I= O
ñ I= N
ó I= O
ó I= P
ó =
=
=
119. iáåÉ~ê=bèì~íáçå=áå=låÉ=s~êá~ÄäÉ=
M
Ä
~ñ =
+ I=
~
Ä
ñ −
= K==
=
120. nì~Çê~íáÅ=bèì~íáçå=
M
Å
Äñ
~ñO
=
+
+ I=
~
O
~Å
Q
Ä
Ä
ñ
O
O
I
N
−
±
−
= K=
=
121. aáëÅêáãáå~åí=
~Å
Q
Ä
a O
−
= =
=
122. sáÉíÉ∞ë=cçêãìä~ë=
fÑ= M
è
éñ
ñO
=
+
+ I=íÜÉå==
=
−
=
+
è
ñ
ñ
é
ñ
ñ
O
N
O
N
K=
=
123. M
Äñ
~ñO
=
+ I= M
ñN = I=
~
Ä
ñO −
= K=
=
124. M
Å
~ñO
=
+ I=
~
Å
ñ O
I
N −
±
= K=
=
125. `ìÄáÅ=bèì~íáçåK=`~êÇ~åç∞ë=cçêãìä~K==
M
è
éó
óP
=
+
+ I==
29. CHAPTER 2. ALGEBRA
19
î
ì
óN +
= I= ( ) ( )á
î
ì
O
P
î
ì
O
N
ó P
I
O +
±
+
−
= I==
ïÜÉêÉ==
P
O
O
P
é
O
è
O
è
ì
+
+
−
= I= P
O
O
P
é
O
è
O
è
î
+
−
−
= K==
=
=
2.7 Inequalities
s~êá~ÄäÉëW=ñI=óI=ò=
oÉ~ä=åìãÄÉêëW=
å
P
O
N ~
I
I
~
I
~
I
~
Ç
I
Å
I
Ä
I
~
K
I=ãI=å=
aÉíÉêãáå~åíëW=aI= ñ
a I= ó
a I= ò
a ==
=
=
126. fåÉèì~äáíáÉëI=fåíÉêî~ä=kçí~íáçåë=~åÇ=dê~éÜë==
=
fåÉèì~äáíó= fåíÉêî~ä=kçí~íáçå= dê~éÜ=
Ä
ñ
~ ≤
≤ = [ ]
Ä
I
~ =
=
Ä
ñ
~ ≤
< = ( ]
Ä
I
~ =
=
Ä
ñ
~ <
≤ = [ )
Ä
I
~ =
=
Ä
ñ
~ <
< = ( )
Ä
I
~ =
=
Ä
ñ ≤
<
∞
− I=
Ä
ñ ≤ =
( ]
Ä
I
∞
− =
=
Ä
ñ <
<
∞
− I=
Ä
ñ < =
( )
Ä
I
∞
− =
=
∞
<
≤ ñ
~ I=
~
ñ ≥ =
[ )
∞
I
~ =
=
∞
<
< ñ
~ I=
~
ñ > =
( )
∞
I
~ =
=
30. CHAPTER 2. ALGEBRA
20
127. fÑ= Ä
~ > I=íÜÉå= ~
Ä < K=
=
128. fÑ= Ä
~ > I=íÜÉå= M
Ä
~ >
− =çê= M
~
Ä <
− K=
=
129. fÑ= Ä
~ > I=íÜÉå= Å
Ä
Å
~ +
>
+ K=
=
130. fÑ= Ä
~ > I=íÜÉå= Å
Ä
Å
~ −
>
− K=
=
131. fÑ= Ä
~ > =~åÇ= Ç
Å > I=íÜÉå= Ç
Ä
Å
~ +
>
+ K=
=
132. fÑ= Ä
~ > =~åÇ= Ç
Å > I=íÜÉå= Å
Ä
Ç
~ −
>
− K=
=
133. fÑ= Ä
~ > =~åÇ= M
ã > I=íÜÉå= ãÄ
ã~ > K=
=
134. fÑ= Ä
~ > =~åÇ= M
ã > I=íÜÉå=
ã
Ä
ã
~
> K=
=
135. fÑ= Ä
~ > =~åÇ= M
ã < I=íÜÉå= ãÄ
ã~ < K=
=
136. fÑ= Ä
~ > =~åÇ= M
ã < I=íÜÉå=
ã
Ä
ã
~
< K=
=
137. fÑ= Ä
~
M <
< =~åÇ= M
å > I=íÜÉå= å
å
Ä
~ < K=
=
138. fÑ= Ä
~
M <
< =~åÇ= M
å < I=íÜÉå= å
å
Ä
~ > K=
=
139. fÑ= Ä
~
M <
< I=íÜÉå= å
å
Ä
~ < K=
=
140.
O
Ä
~
~Ä
+
≤ I==
ïÜÉêÉ= M
~ > =I= M
Ä > X=~å=Éèì~äáíó=áë=î~äáÇ=çåäó=áÑ= Ä
~ = K==
=
141. O
~
N
~ ≥
+ I=ïÜÉêÉ= M
~ > X=~å=Éèì~äáíó=í~âÉë=éä~ÅÉ=çåäó=~í= N
~ = K=
31. CHAPTER 2. ALGEBRA
21
142.
å
~
~
~
~
~
~ å
O
N
å
å
O
N
+
+
+
≤
K
K I=ïÜÉêÉ= M
~
I
I
~
I
~ å
O
N >
K K=
=
143. fÑ= M
Ä
~ñ >
+ =~åÇ= M
~ > I=íÜÉå=
~
Ä
ñ −
> K=
=
144. fÑ= M
Ä
~ñ >
+ =~åÇ= M
~ < I=íÜÉå=
~
Ä
ñ −
< K==
=
145. M
Å
Äñ
~ñO
>
+
+ =
=
= M
~ > = M
~ < =
=
=
=
M
a > =
=
=
N
ñ
ñ < I= O
ñ
ñ > =
=
=
=
O
N ñ
ñ
ñ <
< =
=
=
=
M
a = =
=
ñ
ñN < I= N
ñ
ñ > =
=
=
∅
∈
ñ =
=
=
=
M
a< =
=
=
∞
<
<
∞
− ñ =
=
=
=
∅
∈
ñ =
=
36. CHAPTER 3. GEOMETRY
26
165. ÑÖ
ÜO
= I===
ïÜÉêÉ=Ü=áë=íÜÉ=~äíáíìÇÉ=Ñêçã=íÜÉ=êáÖÜí=~åÖäÉK==
=
166.
Q
~
Ä
ã
O
O
O
~ −
= I=
Q
Ä
~
ã
O
O
O
Ä −
= I===
ïÜÉêÉ= ~
ã =~åÇ= Ä
ã =~êÉ=íÜÉ=ãÉÇá~åë=íç=íÜÉ=äÉÖë=~=~åÇ=ÄK==
=
=
=
Figure 10.
=
167.
O
Å
ãÅ = I==
ïÜÉêÉ= Å
ã =áë=íÜÉ=ãÉÇá~å=íç=íÜÉ=ÜóéçíÉåìëÉ=ÅK=
=
168. Å
ã
O
Å
o =
= =
=
169.
Å
Ä
~
~Ä
O
Å
Ä
~
ê
+
+
=
−
+
= =
=
170. ÅÜ
~Ä = =
=
=
37. CHAPTER 3. GEOMETRY
27
171.
O
ÅÜ
O
~Ä
p =
= =
=
=
=
3.2 Isosceles Triangle
=
_~ëÉW=~=
iÉÖëW=Ä=
_~ëÉ=~åÖäÉW=β =
sÉêíÉñ=~åÖäÉW=α =
^äíáíìÇÉ=íç=íÜÉ=Ä~ëÉW=Ü=
mÉêáãÉíÉêW=i=
^êÉ~W=p=
=
=
=
=
Figure 11.
=
172.
O
VM
α
−
°
=
β =
=
173.
Q
~
Ä
Ü
O
O
O
−
= =
38. CHAPTER 3. GEOMETRY
28
174. Ä
O
~
i +
= =
=
175. α
=
= ëáå
O
Ä
O
~Ü
p
O
=
=
=
=
3.3 Equilateral Triangle
=
páÇÉ=çÑ=~=Éèìáä~íÉê~ä=íêá~åÖäÉW=~=
^äíáíìÇÉW=Ü=
o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o=
o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê=
mÉêáãÉíÉêW=i=
^êÉ~W=p=
=
=
=
=
Figure 12.
=
176.
O
P
~
Ü = =
=
39. CHAPTER 3. GEOMETRY
29
177.
P
P
~
Ü
P
O
o =
= =
=
178.
O
o
S
P
~
Ü
P
N
ê =
=
= =
=
179. ~
P
i = =
=
180.
Q
P
~
O
~Ü
p
O
=
= =
=
=
=
3.4 Scalene Triangle
E^=íêá~åÖäÉ=ïáíÜ=åç=íïç=ëáÇÉë=Éèì~äF=
=
=
páÇÉë=çÑ=~=íêá~åÖäÉW=~I=ÄI=Å=
pÉãáéÉêáãÉíÉêW=
O
Å
Ä
~
é
+
+
= ==
^åÖäÉë=çÑ=~=íêá~åÖäÉW= γ
β
α I
I =
^äíáíìÇÉë=íç=íÜÉ=ëáÇÉë=~I=ÄI=ÅW= Å
Ä
~ Ü
I
Ü
I
Ü =
jÉÇá~åë=íç=íÜÉ=ëáÇÉë=~I=ÄI=ÅW= Å
Ä
~ ã
I
ã
I
ã =
_áëÉÅíçêë=çÑ=íÜÉ=~åÖäÉë= γ
β
α I
I W= Å
Ä
~ í
I
í
I
í =
o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o=
o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê=
^êÉ~W=p=
=
=
40. CHAPTER 3. GEOMETRY
30
===== =
=
Figure 13.
=
181. °
=
γ
+
β
+
α NUM =
=
182. Å
Ä
~ >
+ I==
~
Å
Ä >
+ I==
Ä
Å
~ >
+ K=
=
183. Å
Ä
~ <
− I==
~
Å
Ä <
− I==
Ä
Å
~ <
− K=
=
184. jáÇäáåÉ=
O
~
è = I= ~
öö
è K=
=
===== =
=
Figure 14.
=
41. CHAPTER 3. GEOMETRY
31
185. i~ï=çÑ=`çëáåÉë=
α
−
+
= Åçë
ÄÅ
O
Å
Ä
~ O
O
O
I=
β
−
+
= Åçë
~Å
O
Å
~
Ä O
O
O
I=
γ
−
+
= Åçë
~Ä
O
Ä
~
Å O
O
O
K=
=
186. i~ï=çÑ=páåÉë=
o
O
ëáå
Å
ëáå
Ä
ëáå
~
=
γ
=
β
=
α
I==
ïÜÉêÉ=o=áë=íÜÉ=ê~Çáìë=çÑ=íÜÉ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉK==
=
187.
p
Q
~ÄÅ
Ü
O
~Ä
Ü
O
~Å
Ü
O
ÄÅ
ëáå
O
Å
ëáå
O
Ä
ëáå
O
~
o
Å
Ä
~
=
=
=
=
γ
=
β
=
α
= =
=
188.
( )( )( )
é
Å
é
Ä
é
~
é
êO −
−
−
= I==
Å
Ä
~ Ü
N
Ü
N
Ü
N
ê
N
+
+
= K=
=
189.
( )( )
ÄÅ
Å
é
Ä
é
O
ëáå
−
−
=
α
I=
( )
ÄÅ
~
é
é
O
Åçë
−
=
α
I=
( )( )
( )
~
é
é
Å
é
Ä
é
O
í~å
−
−
−
=
α
K=
=
190. ( )( )( )
Å
é
Ä
é
~
é
é
~
O
Ü~ −
−
−
= I=
( )( )( )
Å
é
Ä
é
~
é
é
Ä
O
ÜÄ −
−
−
= I=
( )( )( )
Å
é
Ä
é
~
é
é
Å
O
ÜÅ −
−
−
= K=
42. CHAPTER 3. GEOMETRY
32
191. β
=
γ
= ëáå
Å
ëáå
Ä
Ü~ I=
α
=
γ
= ëáå
Å
ëáå
~
ÜÄ I=
α
=
β
= ëáå
Ä
ëáå
~
ÜÅ K=
=
192.
Q
~
O
Å
Ä
ã
O
O
O
O
~ −
+
= I==
Q
Ä
O
Å
~
ã
O
O
O
O
Ä −
+
= I==
Q
Å
O
Ä
~
ã
O
O
O
O
Å −
+
= K=
=
===== =
=
Figure 15.
=
193. ~
ã
P
O
^j = I= Ä
ã
P
O
_j = I= Å
ã
P
O
`j = =EcáÖKNRFK=
=
194.
( )
( )O
O
~
Å
Ä
~
é
ÄÅé
Q
í
+
−
= I==
( )
( )O
O
Ä
Å
~
Ä
é
~Åé
Q
í
+
−
= I==
( )
( )O
O
Å
Ä
~
Å
é
~Äé
Q
í
+
−
= K=
=
43. CHAPTER 3. GEOMETRY
33
195.
O
ÅÜ
O
ÄÜ
O
~Ü
p Å
Ä
~
=
=
= I==
O
ëáå
ÄÅ
O
ëáå
~Å
O
ëáå
~Ä
p
α
=
β
=
γ
= I==
( )( )( )
Å
é
Ä
é
~
é
é
p −
−
−
= =EeÉêçå∞ë=cçêãìä~FI=
éê
p = I==
o
Q
~ÄÅ
p = I=
γ
β
α
= ëáå
ëáå
ëáå
o
O
p O
I=
O
í~å
O
í~å
O
í~å
é
p O γ
β
α
= K=
=
=
=
3.5 Square
páÇÉ=çÑ=~=ëèì~êÉW=~=
aá~Öçå~äW=Ç=
o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o=
o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê=
mÉêáãÉíÉêW=i=
^êÉ~W=p=
=
=
=
Figure 16.
44. CHAPTER 3. GEOMETRY
34
196. O
~
Ç = ==
=
197.
O
O
~
O
Ç
o =
= =
=
198.
O
~
ê = =
=
199. ~
Q
i = =
=
200. O
~
p = =
=
=
=
3.6 Rectangle
=
páÇÉë=çÑ=~=êÉÅí~åÖäÉW=~I=Ä=
aá~Öçå~äW=Ç=
o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o=
mÉêáãÉíÉêW=i=
^êÉ~W=p=
=
=
=
=
Figure 17.
=
201. O
O
Ä
~
Ç +
= ==
45. CHAPTER 3. GEOMETRY
35
202.
O
Ç
o = =
=
203. ( )
Ä
~
O
i +
= =
=
204. ~Ä
p = =
=
=
=
3.7 Parallelogram
=
páÇÉë=çÑ=~=é~ê~ääÉäçÖê~ãW=~I=Ä=
aá~Öçå~äëW= O
N Ç
I
Ç =
`çåëÉÅìíáîÉ=~åÖäÉëW= β
αI =
^åÖäÉ=ÄÉíïÉÉå=íÜÉ=Çá~Öçå~äëW=ϕ =
^äíáíìÇÉW=Ü==
mÉêáãÉíÉêW=i=
^êÉ~W=p=
=
=
===== =
=
Figure 18.
=
205. °
=
β
+
α NUM =
=
206. ( )
O
O
O
O
O
N Ä
~
O
Ç
Ç +
=
+ =
=
46. CHAPTER 3. GEOMETRY
36
207. β
=
α
= ëáå
Ä
ëáå
Ä
Ü =
=
208. ( )
Ä
~
O
i +
= =
=
209. α
=
= ëáå
~Ä
~Ü
p I==
ϕ
= ëáå
Ç
Ç
O
N
p O
N K=
=
=
=
3.8 Rhombus
=
páÇÉ=çÑ=~=êÜçãÄìëW=~=
aá~Öçå~äëW= O
N Ç
I
Ç =
`çåëÉÅìíáîÉ=~åÖäÉëW= β
αI =
^äíáíìÇÉW=e=
o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê=
mÉêáãÉíÉêW=i=
^êÉ~W=p=
=
=
===== =
=
Figure 19.
=
47. CHAPTER 3. GEOMETRY
37
210. °
=
β
+
α NUM =
=
211. O
O
O
O
N ~
Q
Ç
Ç =
+ =
=
212.
~
O
Ç
Ç
ëáå
~
Ü O
N
=
α
= =
=
213.
O
ëáå
~
~
Q
Ç
Ç
O
Ü
ê O
N α
=
=
= =
=
214. ~
Q
i = =
=
215. α
=
= ëáå
~
~Ü
p O
I==
O
NÇ
Ç
O
N
p = K=
=
=
=
3.9 Trapezoid
=
_~ëÉë=çÑ=~=íê~éÉòçáÇW=~I=Ä=
jáÇäáåÉW=è=
^äíáíìÇÉW=Ü=
^êÉ~W=p=
=
=
48. CHAPTER 3. GEOMETRY
38
=
=
Figure 20.
=
216.
O
Ä
~
è
+
= =
=
217. èÜ
Ü
O
Ä
~
p =
⋅
+
= =
=
=
=
3.10 Isosceles Trapezoid
=
_~ëÉë=çÑ=~=íê~éÉòçáÇW=~I=Ä=
iÉÖW=Å=
jáÇäáåÉW=è=
^äíáíìÇÉW=Ü=
aá~Öçå~äW=Ç=
o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o=
^êÉ~W=p=
=
=
49. CHAPTER 3. GEOMETRY
39
=
=
Figure 21.
=
218.
O
Ä
~
è
+
= =
=
219. O
Å
~Ä
Ç +
= =
=
220. ( )O
O
~
Ä
Q
N
Å
Ü −
−
= =
=
221.
( )( )
Ä
~
Å
O
Ä
~
Å
O
Å
~Ä
Å
o
O
−
+
+
−
+
= =
=
222. èÜ
Ü
O
Ä
~
p =
⋅
+
= =
=
=
=
=
=
=
50. CHAPTER 3. GEOMETRY
40
3.11 Isosceles Trapezoid with
Inscribed Circle
=
_~ëÉë=çÑ=~=íê~éÉòçáÇW=~I=Ä=
iÉÖW=Å=
jáÇäáåÉW=è=
^äíáíìÇÉW=Ü=
aá~Öçå~äW=Ç=
o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=o=
o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=ê=
mÉêáãÉíÉêW=i=
^êÉ~W=p=
=
=
=
=
Figure 22.
=
223. Å
O
Ä
~ =
+ =
=
224. Å
O
Ä
~
è =
+
= =
=
225. O
O
O
Å
Ü
Ç +
= =
=
51. CHAPTER 3. GEOMETRY
41
226.
O
~Ä
O
Ü
ê =
= =
=
227.
~
Ä
S
Ä
~
U
Ä
~
Å
Ü
Ü
O
Å
~Ä
Å
N
O
Å
ê
Q
ÅÇ
Ü
O
ÅÇ
o O
O
O
+
+
+
=
+
=
+
=
=
= =
=
228. ( ) Å
Q
Ä
~
O
i =
+
= =
=
229.
( )
O
iê
ÅÜ
èÜ
O
~Ä
Ä
~
Ü
O
Ä
~
p =
=
=
+
=
⋅
+
= ==
=
=
=
3.12 Trapezoid with Inscribed Circle
=
_~ëÉë=çÑ=~=íê~éÉòçáÇW=~I=Ä=
i~íÉê~ä=ëáÇÉëW=ÅI=Ç=
jáÇäáåÉW=è=
^äíáíìÇÉW=Ü=
aá~Öçå~äëW= O
N Ç
I
Ç =
^åÖäÉ=ÄÉíïÉÉå=íÜÉ=Çá~Öçå~äëW=ϕ =
o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê=
o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o=
mÉêáãÉíÉêW=i=
^êÉ~W=p=
=
52. CHAPTER 3. GEOMETRY
42
=
=
Figure 23.
=
230. Ç
Å
Ä
~ +
=
+ =
=
231.
O
Ç
Å
O
Ä
~
è
+
=
+
= =
=
232. ( ) ( )
Ç
Å
O
Ä
~
O
i +
=
+
= =
=
233. èÜ
Ü
O
Ç
Å
Ü
O
Ä
~
p =
⋅
+
=
⋅
+
= I==
ϕ
= ëáå
Ç
Ç
O
N
p O
N K=
=
=
=
3.13 Kite
=
páÇÉë=çÑ=~=âáíÉW=~I=Ä=
aá~Öçå~äëW= O
N Ç
I
Ç =
^åÖäÉëW= γ
β
α I
I =
mÉêáãÉíÉêW=i=
^êÉ~W=p=
=
=
53. CHAPTER 3. GEOMETRY
43
=
=
Figure 24.
=
234. °
=
γ
+
β
+
α PSM
O =
=
235. ( )
Ä
~
O
i +
= =
=
236.
O
Ç
Ç
p O
N
= =
=
=
=
3.14 Cyclic Quadrilateral
páÇÉë=çÑ=~=èì~Çêáä~íÉê~äW=~I=ÄI=ÅI=Ç=
aá~Öçå~äëW= O
N Ç
I
Ç =
^åÖäÉ=ÄÉíïÉÉå=íÜÉ=Çá~Öçå~äëW=ϕ =
fåíÉêå~ä=~åÖäÉëW= δ
γ
β
α I
I
I =
o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o=
mÉêáãÉíÉêW=i=
pÉãáéÉêáãÉíÉêW=é==
^êÉ~W=p=
54. CHAPTER 3. GEOMETRY
44
=
=
Figure 25.
=
237. °
=
δ
+
β
=
γ
+
α NUM =
=
238. míçäÉãó∞ë=qÜÉçêÉã=
O
NÇ
Ç
ÄÇ
~Å =
+ =
=
239. Ç
Å
Ä
~
i +
+
+
= =
=
240.
( )( )( )
( )( )( )( )
Ç
é
Å
é
Ä
é
~
é
ÅÇ
~Ä
ÄÅ
~Ç
ÄÇ
~Å
Q
N
o
−
−
−
−
+
+
+
= I==
ïÜÉêÉ=
O
i
é = K=
=
241. ϕ
= ëáå
Ç
Ç
O
N
p O
N I==
( )( )( )( )
Ç
é
Å
é
Ä
é
~
é
p −
−
−
−
= I==
ïÜÉêÉ=
O
i
é = K=
=
=
=
55. CHAPTER 3. GEOMETRY
45
3.15 Tangential Quadrilateral
=
páÇÉë=çÑ=~=èì~Çêáä~íÉê~äW=~I=ÄI=ÅI=Ç=
aá~Öçå~äëW= O
N Ç
I
Ç =
^åÖäÉ=ÄÉíïÉÉå=íÜÉ=Çá~Öçå~äëW=ϕ =
o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê=
mÉêáãÉíÉêW=i=
pÉãáéÉêáãÉíÉêW=é==
^êÉ~W=p=
=
=
=
=
Figure 26.
=
242. Ç
Ä
Å
~ +
=
+ =
=
243. ( ) ( )
Ç
Ä
O
Å
~
O
Ç
Å
Ä
~
i +
=
+
=
+
+
+
= =
=
244.
( ) ( )
é
O
é
Ä
~
Ä
~
Ç
Ç
ê
O
O
O
O
O
N −
+
−
−
= I==
ïÜÉêÉ=
O
i
é = K==
=
56. CHAPTER 3. GEOMETRY
46
245. ϕ
=
= ëáå
Ç
Ç
O
N
éê
p O
N =
=
=
=
3.16 General Quadrilateral
=
páÇÉë=çÑ=~=èì~Çêáä~íÉê~äW=~I=ÄI=ÅI=Ç=
aá~Öçå~äëW= O
N Ç
I
Ç =
^åÖäÉ=ÄÉíïÉÉå=íÜÉ=Çá~Öçå~äëW=ϕ =
fåíÉêå~ä=~åÖäÉëW= δ
γ
β
α I
I
I =
mÉêáãÉíÉêW=i=
^êÉ~W=p=
=
=
======= =
=
Figure 27.
=
246. °
=
δ
+
γ
+
β
+
α PSM =
=
247. Ç
Å
Ä
~
i +
+
+
= =
=
57. CHAPTER 3. GEOMETRY
47
248. ϕ
= ëáå
Ç
Ç
O
N
p O
N =
=
=
=
3.17 Regular Hexagon
=
páÇÉW=~=
fåíÉêå~ä=~åÖäÉW=α =
pä~åí=ÜÉáÖÜíW=ã=
o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê=
o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o=
mÉêáãÉíÉêW=i=
pÉãáéÉêáãÉíÉêW=é==
^êÉ~W=p=
=
=
=
=
Figure 28.
=
249. °
=
α NOM =
=
250.
O
P
~
ã
ê =
= =
58. CHAPTER 3. GEOMETRY
48
251. ~
o = =
=
252. ~
S
i = =
=
253.
O
P
P
~
éê
p
O
=
= I==
ïÜÉêÉ=
O
i
é = K=
=
=
=
3.18 Regular Polygon
=
páÇÉW=~=
kìãÄÉê=çÑ=ëáÇÉëW=å=
fåíÉêå~ä=~åÖäÉW=α =
pä~åí=ÜÉáÖÜíW=ã=
o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê=
o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o=
mÉêáãÉíÉêW=i=
pÉãáéÉêáãÉíÉêW=é==
^êÉ~W=p=
=
=
59. CHAPTER 3. GEOMETRY
49
=
=
Figure 29.
=
254. °
⋅
−
=
α NUM
O
O
å
=
=
255. °
⋅
−
=
α NUM
O
O
å
=
=
256.
å
ëáå
O
~
o
π
= =
=
257.
Q
~
o
å
í~å
O
~
ã
ê
O
O
−
=
π
=
= =
=
258. å~
i = =
=
259.
å
O
ëáå
O
åo
p
O
π
= I==
Q
~
o
é
éê
p
O
O
−
=
= I==
60. CHAPTER 3. GEOMETRY
50
ïÜÉêÉ=
O
i
é = K==
=
=
=
3.19 Circle
=
o~ÇáìëW=o=
aá~ãÉíÉêW=Ç=
`ÜçêÇW=~=
pÉÅ~åí=ëÉÖãÉåíëW=ÉI=Ñ=
q~åÖÉåí=ëÉÖãÉåíW=Ö=
`Éåíê~ä=~åÖäÉW=α =
fåëÅêáÄÉÇ=~åÖäÉW=β =
mÉêáãÉíÉêW=i=
^êÉ~W=p=
=
=
260.
O
ëáå
o
O
~
α
= =
=
=
=
Figure 30.
=
61. CHAPTER 3. GEOMETRY
51
261. O
N
O
N Ä
Ä
~
~ = =
=
=
=
Figure 31.
=
262. N
N ÑÑ
ÉÉ = =
=
===== =
=
Figure 32.
=
263. N
O
ÑÑ
Ö = =
=
62. CHAPTER 3. GEOMETRY
52
===== =
=
Figure 33.
=
264.
O
α
=
β =
=
=
=
Figure 34.
=
265. Ç
o
O
i π
=
π
= =
=
266.
O
io
Q
Ç
o
p
O
O
=
π
=
π
= ==
=
63. CHAPTER 3. GEOMETRY
53
3.20 Sector of a Circle
=
o~Çáìë=çÑ=~=ÅáêÅäÉW=o=
^êÅ=äÉåÖíÜW=ë=
`Éåíê~ä=~åÖäÉ=Eáå=ê~Çá~åëFW=ñ=
`Éåíê~ä=~åÖäÉ=Eáå=ÇÉÖêÉÉëFW=α=
mÉêáãÉíÉêW=i=
^êÉ~W=p=
=
=
=
=
Figure 35.
=
267. oñ
ë = =
=
268.
°
α
π
=
NUM
o
ë =
=
269. o
O
ë
i +
= =
=
270.
°
α
π
=
=
=
PSM
o
O
ñ
o
O
oë
p
O
O
==
=
=
64. CHAPTER 3. GEOMETRY
54
3.21 Segment of a Circle
=
o~Çáìë=çÑ=~=ÅáêÅäÉW=o=
^êÅ=äÉåÖíÜW=ë=
`ÜçêÇW=~=
`Éåíê~ä=~åÖäÉ=Eáå=ê~Çá~åëFW=ñ=
`Éåíê~ä=~åÖäÉ=Eáå=ÇÉÖêÉÉëFW=α=
eÉáÖÜí=çÑ=íÜÉ=ëÉÖãÉåíW=Ü=
mÉêáãÉíÉêW=i=
^êÉ~W=p=
=
=
=
=
Figure 36.
=
271. O
Ü
Üo
O
O
~ −
= =
=
272. O
O
~
o
Q
O
N
o
Ü −
−
= I= o
Ü < =
=
273. ~
ë
i +
= =
=
65. CHAPTER 3. GEOMETRY
55
274. ( )
[ ] ( )
ñ
ëáå
ñ
O
o
ëáå
NUM
O
o
Ü
o
~
ëo
O
N
p
O
O
−
=
α
−
°
απ
=
−
−
= I==
Ü~
P
O
p ≈ K=
=
=
=
3.22 Cube
=
bÇÖÉW=~==
aá~Öçå~äW=Ç=
o~Çáìë=çÑ=áåëÅêáÄÉÇ=ëéÜÉêÉW=ê=
o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ëéÜÉêÉW=ê=
pìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=
=
=== =
=
Figure 37.
=
275. P
~
Ç = =
=
276.
O
~
ê = =
=
66. CHAPTER 3. GEOMETRY
56
277.
O
P
~
o = =
=
278. O
~
S
p = =
=
279. P
~
s = ==
=
=
=
3.23 Rectangular Parallelepiped
=
bÇÖÉëW=~I=ÄI=Å==
aá~Öçå~äW=Ç=
pìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=
=
===== =
=
Figure 38.
=
280. O
O
O
Å
Ä
~
Ç +
+
= =
=
281. ( )
ÄÅ
~Å
~Ä
O
p +
+
= =
=
282. ~ÄÅ
s = ==
67. CHAPTER 3. GEOMETRY
57
3.24 Prism
=
i~íÉê~ä=ÉÇÖÉW=ä=
eÉáÖÜíW=Ü=
i~íÉê~ä=~êÉ~W= i
p =
^êÉ~=çÑ=Ä~ëÉW= _
p =
qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=
=
===== =
=
Figure 39.
=
283. _
i p
O
p
p +
= K==
=
284. i~íÉê~ä=^êÉ~=çÑ=~=oáÖÜí=mêáëã=
( )ä
~
~
~
~
p å
P
O
N
i +
+
+
+
= K =
=
285. i~íÉê~ä=^êÉ~=çÑ=~å=lÄäáèìÉ=mêáëã=
éä
pi = I==
ïÜÉêÉ=é=áë=íÜÉ=éÉêáãÉíÉê=çÑ=íÜÉ=Åêçëë=ëÉÅíáçåK=
=
68. CHAPTER 3. GEOMETRY
58
286. Ü
p
s _
= =
=
287. `~î~äáÉêáDë=mêáåÅáéäÉ==
dáîÉå=íïç=ëçäáÇë=áåÅäìÇÉÇ=ÄÉíïÉÉå=é~ê~ääÉä=éä~åÉëK=fÑ=ÉîÉêó=
éä~åÉ=Åêçëë=ëÉÅíáçå=é~ê~ääÉä=íç=íÜÉ=ÖáîÉå=éä~åÉë=Ü~ë=íÜÉ=ë~ãÉ=
~êÉ~=áå=ÄçíÜ=ëçäáÇëI=íÜÉå=íÜÉ=îçäìãÉë=çÑ=íÜÉ=ëçäáÇë=~êÉ=Éèì~äK=
=
=
=
3.25 Regular Tetrahedron
=
qêá~åÖäÉ=ëáÇÉ=äÉåÖíÜW=~=
eÉáÖÜíW=Ü=
^êÉ~=çÑ=Ä~ëÉW= _
p =
pìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=
=
=
=
Figure 40.
=
288. ~
P
O
Ü = =
=
69. CHAPTER 3. GEOMETRY
59
289.
Q
~
P
p
O
_ = =
=
290. O
~
P
p = =
=
291.
O
S
~
Ü
p
P
N
s
P
_ =
= K==
=
=
=
3.26 Regular Pyramid
=
páÇÉ=çÑ=Ä~ëÉW=~=
i~íÉê~ä=ÉÇÖÉW=Ä=
eÉáÖÜíW=Ü=
pä~åí=ÜÉáÖÜíW=ã==
kìãÄÉê=çÑ=ëáÇÉëW=å==
pÉãáéÉêáãÉíÉê=çÑ=Ä~ëÉW=é=
o~Çáìë=çÑ=áåëÅêáÄÉÇ=ëéÜÉêÉ=çÑ=Ä~ëÉW=ê=
^êÉ~=çÑ=Ä~ëÉW= _
p =
i~íÉê~ä=ëìêÑ~ÅÉ=~êÉ~W= i
p =
qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=
=
70. CHAPTER 3. GEOMETRY
60
=
=
Figure 41.
=
292.
Q
~
Ä
ã
O
O
−
= =
=
293.
å
ëáå
O
~
å
ëáå
Ä
Q
Ü
O
O
O
π
−
π
= =
=
294. éã
~
Ä
Q
å~
Q
N
å~ã
O
N
p O
O
i =
−
=
= =
=
295. éê
p_ = =
=
296. i
_ p
p
p +
= =
=
297. éêÜ
P
N
Ü
p
P
N
s _ =
= ==
=
=
=
71. CHAPTER 3. GEOMETRY
61
3.27 Frustum of a Regular Pyramid
=
_~ëÉ=~åÇ=íçé=ëáÇÉ=äÉåÖíÜëW=
å
P
O
N
å
P
O
N
Ä
I
I
Ä
I
Ä
I
Ä
~
I
I
~
I
~
I
~
K
K
=
eÉáÖÜíW=Ü=
pä~åí=ÜÉáÖÜíW=ã==
^êÉ~=çÑ=Ä~ëÉëW= N
p I= O
p =
i~íÉê~ä=ëìêÑ~ÅÉ=~êÉ~W= i
p =
mÉêáãÉíÉê=çÑ=Ä~ëÉëW= N
m I= O
m =
pÅ~äÉ=Ñ~ÅíçêW=â=
qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=
=
=
=
Figure 42.
=
298. â
~
Ä
~
Ä
~
Ä
~
Ä
~
Ä
å
å
P
P
O
O
N
N
=
=
=
=
=
= K =
=
72. CHAPTER 3. GEOMETRY
62
299. O
N
O
â
p
p
= =
=
300.
( )
O
m
m
ã
p O
N
i
+
= =
=
301. O
N
i p
p
p
p +
+
= =
=
302. ( )
O
O
N
N p
p
p
p
P
Ü
s +
+
= =
=
303. [ ]
O
N
O
N
â
â
N
P
Üp
~
Ä
~
Ä
N
P
Üp
s +
+
=
+
+
= =
=
=
=
3.28 Rectangular Right Wedge
=
páÇÉë=çÑ=Ä~ëÉW=~I=Ä=
qçé=ÉÇÖÉW=Å=
eÉáÖÜíW=Ü=
i~íÉê~ä=ëìêÑ~ÅÉ=~êÉ~W= i
p =
^êÉ~=çÑ=Ä~ëÉW= _
p =
qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=
=
73. CHAPTER 3. GEOMETRY
63
=
=
Figure 43.
=
304. ( ) ( )O
O
O
O
i Å
~
Ü
Ä
Ä
Ü
Q
Å
~
O
N
p −
+
+
+
+
= =
=
305. ~Ä
p_ = =
=
306. i
_ p
p
p +
= =
=
307. ( )
Å
~
O
S
ÄÜ
s +
= =
=
=
=
3.29 Platonic Solids
=
bÇÖÉW=~=
o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê=
o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o=
pìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=
=
75. CHAPTER 3. GEOMETRY
65
311. P
~
O
p O
= =
=
312.
P
O
~
s
P
= =
=
=
Icosahedron
=
=
=
Figure 45.
=
313.
( )
NO
R
P
P
~
ê
+
= =
=
314. ( )
R
R
O
Q
~
o +
= =
=
315. P
~
R
p O
= =
=
316.
( )
NO
R
P
~
R
s
P
+
= =
=
=
76. CHAPTER 3. GEOMETRY
66
Dodecahedron
=
=
=
Figure 46.
=
317.
( )
O
R
NN
OR
NM
~
ê
+
= =
=
318.
( )
Q
R
N
P
~
o
+
= =
=
319. ( )
R
O
R
R
~
P
p O
+
= =
=
320.
( )
Q
R
T
NR
~
s
P
+
= =
=
=
=
3.30 Right Circular Cylinder
=
o~Çáìë=çÑ=Ä~ëÉW=o=
aá~ãÉíÉê=çÑ=Ä~ëÉW=Ç=
77. CHAPTER 3. GEOMETRY
67
eÉáÖÜíW=e=
i~íÉê~ä=ëìêÑ~ÅÉ=~êÉ~W= i
p =
^êÉ~=çÑ=Ä~ëÉW= _
p =
qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=
=
===== =
=
Figure 47.
=
321. oe
O
pi π
= =
=
322. ( )
+
π
=
+
π
=
+
=
O
Ç
e
Ç
o
e
o
O
p
O
p
p _
i =
=
323. e
o
e
p
s O
_ π
=
= =
=
=
=
78. CHAPTER 3. GEOMETRY
68
3.31 Right Circular Cylinder with
an Oblique Plane Face
=
o~Çáìë=çÑ=Ä~ëÉW=o=
qÜÉ=ÖêÉ~íÉëí=ÜÉáÖÜí=çÑ=~=ëáÇÉW= N
Ü =
qÜÉ=ëÜçêíÉëí=ÜÉáÖÜí=çÑ=~=ëáÇÉW= O
Ü =
i~íÉê~ä=ëìêÑ~ÅÉ=~êÉ~W= i
p =
^êÉ~=çÑ=éä~åÉ=ÉåÇ=Ñ~ÅÉëW= _
p =
qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=
=
=
=
Figure 48.
=
324. ( )
O
N
i Ü
Ü
o
p +
π
= =
=
325.
O
O
N
O
O
_
O
Ü
Ü
o
o
o
p
−
+
π
+
π
= =
=
79. CHAPTER 3. GEOMETRY
69
326.
−
+
+
+
+
π
=
+
=
O
O
N
O
O
N
_
i
O
Ü
Ü
o
o
Ü
Ü
o
p
p
p =
=
327. ( )
O
N
O
Ü
Ü
O
o
s +
π
= =
=
=
=
3.32 Right Circular Cone
o~Çáìë=çÑ=Ä~ëÉW=o=
aá~ãÉíÉê=çÑ=Ä~ëÉW=Ç=
eÉáÖÜíW=e=
pä~åí=ÜÉáÖÜíW=ã=
i~íÉê~ä=ëìêÑ~ÅÉ=~êÉ~W= i
p =
^êÉ~=çÑ=Ä~ëÉW= _
p =
qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=
=
=
=
Figure 49.
80. CHAPTER 3. GEOMETRY
70
328. O
O
o
ã
e −
= =
=
329.
O
ãÇ
oã
pi
π
=
π
= =
=
330. O
_ o
p π
= =
=
331. ( )
+
π
=
+
π
=
+
=
O
Ç
ã
Ç
O
N
o
ã
o
p
p
p _
i =
=
332. e
o
P
N
e
p
P
N
s O
_ π
=
= =
=
=
=
3.33 Frustum of a Right Circular Cone
=
o~Çáìë=çÑ=Ä~ëÉëW=oI=ê=
eÉáÖÜíW=e=
pä~åí=ÜÉáÖÜíW=ã=
pÅ~äÉ=Ñ~ÅíçêW=â=
^êÉ~=çÑ=Ä~ëÉëW= N
p I= O
p =
i~íÉê~ä=ëìêÑ~ÅÉ=~êÉ~W= i
p =
qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=
=
81. CHAPTER 3. GEOMETRY
71
=
=
Figure 50.
=
333. ( )O
O
ê
o
ã
e −
−
= =
=
334. â
ê
o
= =
=
335. O
O
O
N
O
â
ê
o
p
p
=
= =
=
336. ( )
ê
o
ã
pi +
π
= =
=
337. ( )
[ ]
ê
o
ã
ê
o
p
p
p
p O
O
i
O
N +
+
+
π
=
+
+
= =
=
338. ( )
O
O
N
N p
p
p
p
P
Ü
s +
+
= =
=
339. [ ]
O
N
O
N
â
â
N
P
Üp
ê
o
ê
o
N
P
Üp
s +
+
=
+
+
= =
=
=
=
82. CHAPTER 3. GEOMETRY
72
3.34 Sphere
=
o~ÇáìëW=o=
aá~ãÉíÉêW=Ç=
pìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=
=
=
Figure 51.
=
340. O
o
Q
p π
= =
=
341. po
P
N
Ç
S
N
e
o
P
Q
s P
P
=
π
=
π
= =
=
=
=
3.35 Spherical Cap
o~Çáìë=çÑ=ëéÜÉêÉW=o=
o~Çáìë=çÑ=Ä~ëÉW=ê=
eÉáÖÜíW=Ü=
^êÉ~=çÑ=éä~åÉ=Ñ~ÅÉW= _
p =
^êÉ~=çÑ=ëéÜÉêáÅ~ä=Å~éW= `
p =
qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
83. CHAPTER 3. GEOMETRY
73
=
=
Figure 52.
=
342.
Ü
O
Ü
ê
o
O
O
+
= =
=
343. O
_ ê
p π
= =
=
344. ( )
O
O
` ê
Ü
p +
π
= =
=
345. ( ) ( )
O
O
O
`
_ ê
oÜ
O
ê
O
Ü
p
p
p +
π
=
+
π
=
+
= =
=
346. ( ) ( )
O
O
O
Ü
ê
P
Ü
S
Ü
o
P
Ü
S
s +
π
=
−
π
= =
=
=
=
3.36 Spherical Sector
=
o~Çáìë=çÑ=ëéÜÉêÉW=o=
o~Çáìë=çÑ=Ä~ëÉ=çÑ=ëéÜÉêáÅ~ä=Å~éW=ê=
eÉáÖÜíW=Ü=
qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=
84. CHAPTER 3. GEOMETRY
74
====== === =
=
Figure 53.
=
347. ( )
ê
Ü
O
o
p +
π
= =
=
348. Ü
o
P
O
s O
π
= =
=
kçíÉW=qÜÉ=ÖáîÉå=Ñçêãìä~ë=~êÉ=ÅçêêÉÅí=ÄçíÜ=Ñçê=±çéÉå≤=~åÇ=
±ÅäçëÉÇ≤=ëéÜÉêáÅ~ä=ëÉÅíçêK=
=
=
=
3.37 Spherical Segment
=
o~Çáìë=çÑ=ëéÜÉêÉW=o=
o~Çáìë=çÑ=Ä~ëÉëW= N
ê I= O
ê =
eÉáÖÜíW=Ü=
^êÉ~=çÑ=ëéÜÉêáÅ~ä=ëìêÑ~ÅÉW= p
p =
^êÉ~=çÑ=éä~åÉ=ÉåÇ=Ñ~ÅÉëW= N
p I= O
p =
qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=
85. CHAPTER 3. GEOMETRY
75
===== =
=
Figure 54.
=
349. oÜ
O
pp π
= =
=
350. ( )
O
O
O
N
O
N
p ê
ê
oÜ
O
p
p
p
p +
+
π
=
+
+
= =
=
351. ( )
O
O
O
O
N Ü
ê
P
ê
P
Ü
S
N
s +
+
π
= =
=
=
=
3.38 Spherical Wedge
=
o~ÇáìëW=o=
aáÜÉÇê~ä=~åÖäÉ=áå=ÇÉÖêÉÉëW=ñ=
aáÜÉÇê~ä=~åÖäÉ=áå=ê~Çá~åëW=α=
^êÉ~=çÑ=ëéÜÉêáÅ~ä=äìåÉW= i
p =
qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=
=
86. CHAPTER 3. GEOMETRY
76
=
=
Figure 55.
=
352. ñ
o
O
VM
o
p O
O
i =
α
π
= =
=
353. ñ
o
O
o
VM
o
o
p O
O
O
O
+
π
=
α
π
+
π
= =
=
354. ñ
o
P
O
OTM
o
s P
P
=
α
π
= =
=
=
=
3.39 Ellipsoid
=
pÉãá-~ñÉëW=~I=ÄI=Å=
sçäìãÉW=s=
87. CHAPTER 3. GEOMETRY
77
======= =
=
Figure 56.
=
355. ~ÄÅ
P
Q
s π
= =
=
=
=
Prolate Spheroid
=
pÉãá-~ñÉëW=~I=ÄI=Ä=E Ä
~ > F=
pìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=
=
356.
+
π
=
É
É
~êÅëáå
~
Ä
Ä
O
p I==
ïÜÉêÉ=
~
Ä
~
É
O
O
−
= K=
=
357. ~
Ä
P
Q
s O
π
= =
=
88. CHAPTER 3. GEOMETRY
78
Oblate Spheroid
=
pÉãá-~ñÉëW=~I=ÄI=Ä=E Ä
~ < F=
pìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=
=
358.
+
π
=
~
L
ÄÉ
~
ÄÉ
~êÅëáåÜ
~
Ä
Ä
O
p I==
ïÜÉêÉ=
Ä
~
Ä
É
O
O
−
= K=
=
359. ~
Ä
P
Q
s O
π
= =
=
=
=
3.40 Circular Torus
=
j~àçê=ê~ÇáìëW=o=
jáåçê=ê~ÇáìëW=ê=
pìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=
90. 80
Chapter 4
Trigonometry
=
=
=
=
^åÖäÉëW=α I=β =
oÉ~ä=åìãÄÉêë=EÅççêÇáå~íÉë=çÑ=~=éçáåíFW=ñI=ó==
tÜçäÉ=åìãÄÉêW=â=
=
=
4.1 Radian and Degree Measures of Angles
=
362. ?
QR
D
NT
RT
NUM
ê~Ç
N °
≈
π
°
= =
=
363. ê~Ç
MNTQRP
K
M
ê~Ç
NUM
N ≈
π
=
° =
=
364. ê~Ç
MMMOVN
K
M
ê~Ç
SM
NUM
D
N ≈
⋅
π
= =
=
365. ê~Ç
MMMMMR
K
M
ê~Ç
PSMM
NUM
?
N ≈
⋅
π
= =
=
366. =
=
^åÖäÉ=
EÇÉÖêÉÉëF=
M= PM= QR= SM= VM= NUM= OTM= PSM=
^åÖäÉ=
Eê~Çá~åëF= M=
S
π
=
Q
π
=
P
π
=
O
π
= π=
O
Pπ
= π
O =
=
=
=
91. CHAPTER 4. TRIGONOMETRY
81
4.2 Definitions and Graphs of Trigonometric
Functions
=
= =
=
Figure 58.
=
367.
ê
ó
ëáå =
α =
=
368.
ê
ñ
Åçë =
α =
=
369.
ñ
ó
í~å =
α =
=
370.
ó
ñ
Åçí =
α =
=
92. CHAPTER 4. TRIGONOMETRY
82
371.
ñ
ê
ëÉÅ =
α =
=
372.
ó
ê
ÅçëÉÅ =
α =
=
373. páåÉ=cìåÅíáçå=
ñ
ëáå
ó = I= N
ñ
ëáå
N ≤
≤
− K=
=
=
Figure 59.
=
374. `çëáåÉ=cìåÅíáçå==
ñ
Åçë
ó = I= N
ñ
Åçë
N ≤
≤
− K=
97. CHAPTER 4. TRIGONOMETRY
87
4.4 Trigonometric Functions of Common
Angles
381. =
°
α = ê~Ç
α = α
ëáå = α
Åçë = α
í~å = α
Åçí α
ëÉÅ = α
ÅçëÉÅ =
M= M= M= N= M= ∞= N= ∞=
PM=
S
π
=
O
N
=
O
P
=
P
N
= P =
P
O
= O=
QR=
Q
π
=
O
O
=
O
O
= N= N= O = O =
SM=
P
π
=
O
P
=
O
N
= P =
P
N
= O=
P
O
=
VM=
O
π
= N= M= ∞ = M= ∞ = N=
NOM=
P
Oπ
=
O
P
=
O
N
− = P
− =
P
N
− O
− =
P
O
=
NUM= π= M= N
− = M= ∞ = N
− = ∞ =
OTM=
O
Pπ
= N
− = M= ∞= M= ∞= N
− =
PSM= π
O = M= N= M= ∞ = N= ∞ =
=
=
=
=
=
=
=
=
=
=
=
=
=
98. CHAPTER 4. TRIGONOMETRY
88
382. =
°
α = ê~Ç
α = α
ëáå = α
Åçë = α
í~å = α
Åçí =
NR=
NO
π
=
Q
O
S −
=
Q
O
S +
= P
O− = P
O+ =
NU=
NM
π
=
Q
N
R −
=
Q
R
O
NM +
R
R
O
R−
= R
O
R+ =
PS=
R
π
=
Q
R
O
NM −
Q
N
R +
=
N
R
R
O
NM
+
−
R
O
NM
N
R
−
+
=
RQ=
NM
Pπ
=
Q
N
R +
=
Q
R
O
NM −
R
O
NM
N
R
−
+
N
R
R
O
NM
+
−
=
TO=
R
Oπ
=
Q
R
O
NM +
Q
N
R −
= R
O
R+ = R
R
O
R−
=
TR=
NO
Rπ
=
Q
O
S +
=
Q
O
S −
= P
O+ = P
O− =
=
=
=
4.5 Most Important Formulas
=
383. N
Åçë
ëáå O
O
=
α
+
α =
=
384. N
í~å
ëÉÅ O
O
=
α
−
α =
=
385. N
Åçí
ÅëÅ O
O
=
α
−
α =
=
386.
α
α
=
α
Åçë
ëáå
í~å =
100. CHAPTER 4. TRIGONOMETRY
90
4.7 Periodicity of Trigonometric Functions
=
392. ( ) α
=
π
±
α ëáå
å
O
ëáå I=éÉêáçÇ= π
O =çê= °
PSM K=
=
393. ( ) α
=
π
±
α Åçë
å
O
Åçë I=éÉêáçÇ= π
O =çê= °
PSM K=
=
394. ( ) α
=
π
±
α í~å
å
í~å I=éÉêáçÇ=π=çê= °
NUM K=
=
395. ( ) α
=
π
±
α Åçí
å
Åçí I=éÉêáçÇ=π=çê= °
NUM K=
=
=
=
4.8 Relations between Trigonometric
Functions
=
396. ( ) N
Q
O
Åçë
O
O
Åçë
N
O
N
Åçë
N
ëáå O
O
−
π
−
α
=
α
−
±
=
α
−
±
=
α =
=
O
í~å
N
O
í~å
O
O α
+
α
= =
=
397. ( ) N
O
Åçë
O
O
Åçë
N
O
N
ëáå
N
Åçë O
O
−
α
=
α
+
±
=
α
−
±
=
α =
=
O
í~å
N
O
í~å
N
O
O
α
+
α
−
= =
=
398.
α
α
−
=
α
+
α
=
−
α
±
=
α
α
=
α
O
ëáå
O
Åçë
N
O
Åçë
N
O
ëáå
N
ëÉÅ
Åçë
ëáå
í~å O
=
101. CHAPTER 4. TRIGONOMETRY
91
=
O
í~å
N
O
í~å
O
O
Åçë
N
O
Åçë
N
O α
+
α
=
α
+
α
−
±
= =
=
399.
α
−
α
=
α
α
+
=
−
α
±
=
α
α
=
α
O
Åçë
N
O
ëáå
O
ëáå
O
Åçë
N
N
ÅëÅ
ëáå
Åçë
Åçí O
=
=
O
í~å
O
O
í~å
N
O
Åçë
N
O
Åçë
N
O
α
α
−
=
α
−
α
+
±
= =
=
400.
O
í~å
N
O
í~å
N
í~å
N
Åçë
N
ëÉÅ
O
O
O
α
−
α
+
=
α
+
±
=
α
=
α =
=
401.
O
í~å
O
O
í~å
N
Åçí
N
ëáå
N
ÅëÅ
O
O
α
α
+
=
α
+
±
=
α
=
α =
=
=
=
4.9 Addition and Subtraction Formulas
=
402. ( ) α
β
+
β
α
=
β
+
α Åçë
ëáå
Åçë
ëáå
ëáå =
=
403. ( ) α
β
−
β
α
=
−
α Åçë
ëáå
Åçë
ëáå
ó
ëáå =
=
404. ( ) β
α
−
β
α
=
β
+
α ëáå
ëáå
Åçë
Åçë
Åçë =
=
405. ( ) β
α
+
β
α
=
β
−
α ëáå
ëáå
Åçë
Åçë
Åçë =
102. CHAPTER 4. TRIGONOMETRY
92
406. ( )
β
α
−
β
+
α
=
β
+
α
í~å
í~å
N
í~å
í~å
í~å =
=
407. ( )
β
α
+
β
−
α
=
β
−
α
í~å
í~å
N
í~å
í~å
í~å =
=
408. ( )
β
+
α
β
α
−
=
β
+
α
í~å
í~å
í~å
í~å
N
Åçí =
=
409. ( )
β
−
α
β
α
+
=
β
−
α
í~å
í~å
í~å
í~å
N
Åçí =
=
=
=
4.10 Double Angle Formulas
=
410. α
⋅
α
=
α Åçë
ëáå
O
O
ëáå =
=
411. N
Åçë
O
ëáå
O
N
ëáå
Åçë
O
Åçë O
O
O
O
−
α
=
α
−
=
α
−
α
=
α =
=
412.
α
−
α
=
α
−
α
=
α
í~å
Åçí
O
í~å
N
í~å
O
O
í~å O
=
=
413.
O
í~å
Åçí
Åçí
O
N
Åçí
O
Åçí
O
α
−
α
=
α
−
α
=
α =
=
=
=
=
=
=
103. CHAPTER 4. TRIGONOMETRY
93
4.11 Multiple Angle Formulas
=
414. α
−
α
⋅
α
=
α
−
α
=
α P
O
P
ëáå
ëáå
Åçë
P
ëáå
Q
ëáå
P
P
ëáå =
=
415. α
⋅
α
−
α
⋅
α
=
α Åçë
ëáå
U
Åçë
ëáå
Q
Q
ëáå P
=
=
416. α
+
α
−
α
=
α R
P
ëáå
NS
ëáå
OM
ëáå
R
R
ëáå =
=
417. α
⋅
α
−
α
=
α
−
α
=
α O
P
P
ëáå
Åçë
P
Åçë
Åçë
P
Åçë
Q
P
Åçë =
=
418. N
Åçë
U
Åçë
U
Q
Åçë O
Q
+
α
−
α
=
α =
=
419. α
+
α
−
α
=
α Åçë
R
Åçë
OM
Åçë
NS
R
Åçë P
R
=
=
420.
α
−
α
−
α
=
α O
P
í~å
P
N
í~å
í~å
P
P
í~å =
=
421.
α
+
α
−
α
−
α
=
α Q
O
P
í~å
í~å
S
N
í~å
Q
í~å
Q
Q
í~å =
=
422.
α
+
α
−
α
+
α
−
α
=
α Q
O
P
R
í~å
R
í~å
NM
N
í~å
R
í~å
NM
í~å
R
í~å =
=
423.
N
Åçí
P
Åçí
P
Åçí
P
Åçí O
P
−
α
α
−
α
=
α =
=
424.
α
−
α
α
+
α
−
=
α P
Q
O
í~å
Q
í~å
Q
í~å
í~å
S
N
Q
Åçí ==
=
104. CHAPTER 4. TRIGONOMETRY
94
425.
α
+
α
−
α
α
+
α
−
=
α
í~å
R
í~å
NM
í~å
í~å
R
í~å
NM
N
R
Åçí P
R
Q
O
=
=
=
=
4.12 Half Angle Formulas
=
426.
O
Åçë
N
O
ëáå
α
−
±
=
α
=
=
427.
O
Åçë
N
O
Åçë
α
+
±
=
α
=
=
428. α
−
α
=
α
α
−
=
α
+
α
=
α
+
α
−
±
=
α
Åçí
ÅëÅ
ëáå
Åçë
N
Åçë
N
ëáå
Åçë
N
Åçë
N
O
í~å =
=
429. α
+
α
=
α
α
+
=
α
−
α
=
α
−
α
+
±
=
α
Åçí
ÅëÅ
ëáå
Åçë
N
Åçë
N
ëáå
Åçë
N
Åçë
N
O
Åçí =
=
=
=
4.13 Half Angle Tangent Identities
=
430.
O
í~å
N
O
í~å
O
ëáå
O α
+
α
=
α =
=
105. CHAPTER 4. TRIGONOMETRY
95
431.
O
í~å
N
O
í~å
N
Åçë
O
O
α
+
α
−
=
α =
=
432.
O
í~å
N
O
í~å
O
í~å
O α
−
α
=
α =
=
433.
O
í~å
O
O
í~å
N
Åçí
O
α
α
−
=
α =
=
=
=
4.14 Transforming of Trigonometric
Expressions to Product
=
434.
O
Åçë
O
ëáå
O
ëáå
ëáå
β
−
α
β
+
α
=
β
+
α =
=
435.
O
ëáå
O
Åçë
O
ëáå
ëáå
β
−
α
β
+
α
=
β
−
α =
=
436.
O
Åçë
O
Åçë
O
Åçë
Åçë
β
−
α
β
+
α
=
β
+
α =
=
437.
O
ëáå
O
ëáå
O
Åçë
Åçë
β
−
α
β
+
α
−
=
β
−
α =
=
111. CHAPTER 4. TRIGONOMETRY
101
469. fåîÉêëÉ=`çí~åÖÉåí=cìåÅíáçå==
ñ
Åçí
~êÅ
ó = I= ∞
≤
≤
∞
− ñ I= π
<
< ñ
Åçí
~êÅ
M K=
===== =
Figure 69.
=
470. fåîÉêëÉ=pÉÅ~åí=cìåÅíáçå==
( ] [ ) K
I
O
O
I
M
ñ
ëÉÅ
~êÅ
I
I
N
N
I
ñ
I
ñ
=
~êÅëÉÅ
ó
π
π
∪
π
∈
∞
∪
−
∞
−
∈
=
=
Figure 70.
112. CHAPTER 4. TRIGONOMETRY
102
471. fåîÉêëÉ=`çëÉÅ~åí=cìåÅíáçå==
( ] [ ) K
O
I
M
M
I
O
ñ
ÅëÅ
~êÅ
I
I
N
N
I
ñ
I
ñ
~êÅÅëÅ
ó
π
∪
π
−
∈
∞
∪
−
∞
−
∈
=
=
=
Figure 71.
=
=
4.18 Principal Values of Inverse
Trigonometric Functions
472.
ñ = M=
O
N
=
O
O
=
O
P
N=
ñ
~êÅëáå = °
M = °
PM = °
QR = °
SM °
VM
ñ
~êÅÅçë = °
VM °
SM = °
QR = °
PM °
M =
ñ = O
N
−
O
O
−
O
P
− N
− = =
ñ
~êÅëáå =
°
−PM
=
°
− QR °
− SM
°
− VM
=
=
ñ
~êÅÅçë =
°
NOM
=
°
NPR = °
NRM =
°
NUM
=
=
113. CHAPTER 4. TRIGONOMETRY
103
473.
ñ = M=
P
P
N= P =
P
P
− N
− = P
− =
ñ
~êÅí~å = °
M = °
PM °
QR °
SM °
−PM
°
− QR
=
°
− SM =
ñ
Åçí
~êÅ = °
VM °
SM °
QR °
PM °
NOM =
°
NPR
=
°
NRM =
=
=
=
4.19 Relations between Inverse
Trigonometric Functions
=
474. ( ) ñ
~êÅëáå
ñ
~êÅëáå −
=
− =
=
475. ñ
~êÅÅçë
O
ñ
~êÅëáå −
π
= =
=
476. O
ñ
N
~êÅÅçë
ñ
~êÅëáå −
= I= N
ñ
M ≤
≤ K=
=
477. O
ñ
N
~êÅÅçë
ñ
~êÅëáå −
−
= I= M
ñ
N ≤
≤
− K=
=
478.
O
ñ
N
ñ
~êÅí~å
ñ
~êÅëáå
−
= I= N
ñO
< K=
=
479.
ñ
ñ
N
Åçí
~êÅ
ñ
~êÅëáå
O
−
= I= N
ñ
M ≤
< K=
=
480. π
−
−
=
ñ
ñ
N
Åçí
~êÅ
ñ
~êÅëáå
O
I= M
ñ
N <
≤
− K=
=
481. ( ) ñ
~êÅÅçë
ñ
~êÅÅçë −
π
=
− =
114. CHAPTER 4. TRIGONOMETRY
104
482. ñ
~êÅëáå
O
ñ
~êÅÅçë −
π
= =
=
483. O
ñ
N
~êÅëáå
ñ
~êÅÅçë −
= I= N
ñ
M ≤
≤ K=
=
484. O
ñ
N
~êÅëáå
ñ
~êÅÅçë −
−
π
= I= M
ñ
N ≤
≤
− K=
=
485.
ñ
ñ
N
~êÅí~å
ñ
~êÅÅçë
O
−
= I= N
ñ
M ≤
< K=
=
486.
ñ
ñ
N
~êÅí~å
ñ
~êÅÅçë
O
−
+
π
= I= M
ñ
N <
≤
− K=
=
487.
O
ñ
N
ñ
Åçí
~êÅ
ñ
~êÅÅçë
−
= I= N
ñ
N ≤
≤
− K=
=
488. ( ) ñ
~êÅí~å
ñ
~êÅí~å −
=
− =
=
489. ñ
Åçí
~êÅ
O
ñ
~êÅí~å −
π
= =
=
490.
O
ñ
N
ñ
~êÅëáå
ñ
~êÅí~å
+
= =
=
491.
O
ñ
N
N
~êÅÅçë
ñ
~êÅí~å
+
= I= M
ñ ≥ K=
=
492.
O
ñ
N
N
~êÅÅçë
ñ
~êÅí~å
+
−
= I= M
ñ ≤ K=
=
115. CHAPTER 4. TRIGONOMETRY
105
493.
ñ
N
~êÅí~å
O
ñ
~êÅí~å −
π
= I= M
ñ > K=
=
494.
ñ
N
~êÅí~å
O
ñ
~êÅí~å −
π
−
= I= M
ñ < K=
=
495.
ñ
N
Åçí
~êÅ
ñ
~êÅí~å = I= M
ñ > K=
=
496. π
−
=
ñ
N
Åçí
~êÅ
ñ
~êÅí~å I= M
ñ < K=
=
497. ( ) ñ
Åçí
~êÅ
ñ
Åçí
~êÅ −
π
=
− =
=
498. ñ
~êÅí~å
O
ñ
Åçí
~êÅ −
π
= =
=
499.
O
ñ
N
N
~êÅëáå
ñ
Åçí
~êÅ
+
= I= M
ñ > K=
=
500.
O
ñ
N
N
~êÅëáå
ñ
Åçí
~êÅ
+
−
π
= I= M
ñ < K=
=
501.
O
ñ
N
ñ
~êÅÅçë
ñ
Åçí
~êÅ
+
= =
=
502.
ñ
N
~êÅí~å
ñ
Åçí
~êÅ = I= M
ñ > K=
=
503.
ñ
N
~êÅí~å
ñ
Åçí
~êÅ +
π
= I= M
ñ < K=
=
=
117. 107
Chapter 5
Matrices and Determinants
=
=
=
=
j~íêáÅÉëW=^I=_I=`=
bäÉãÉåíë=çÑ=~=ã~íêáñW= á
~ I= á
Ä I= áà
~ I= áà
Ä I= áà
Å =
aÉíÉêãáå~åí=çÑ=~=ã~íêáñW= ^
ÇÉí =
jáåçê=çÑ=~å=ÉäÉãÉåí= áà
~ W= áà
j =
`çÑ~Åíçê=çÑ=~å=ÉäÉãÉåí= áà
~ W= áà
` =
qê~åëéçëÉ=çÑ=~=ã~íêáñW= q
^ I= ^
ú
=
^Çàçáåí=çÑ=~=ã~íêáñW= ^
~Çà =
qê~ÅÉ=çÑ=~=ã~íêáñW= ^
íê =
fåîÉêëÉ=çÑ=~=ã~íêáñW= N
^−
=
oÉ~ä=åìãÄÉêW=â=
oÉ~ä=î~êá~ÄäÉëW= á
ñ =
k~íìê~ä=åìãÄÉêëW=ãI=å===
=
=
5.1 Determinants
=
513. pÉÅçåÇ=lêÇÉê=aÉíÉêãáå~åí=
N
O
O
N
O
O
N
N
Ä
~
Ä
~
Ä
~
Ä
~
^
ÇÉí −
=
= =
=
=
=
=
=
118. CHAPTER 5. MATRICES AND DETERMINANTS
108
514. qÜáêÇ=lêÇÉê=aÉíÉêãáå~åí=
−
+
+
=
= PO
ON
NP
PN
OP
NO
PP
OO
NN
PP
PO
PN
OP
OO
ON
NP
NO
NN
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
^
ÇÉí =
PN
OO
NP
PP
ON
NO
PO
OP
NN ~
~
~
~
~
~
~
~
~ −
−
− =
=
515. p~êêìë=oìäÉ=E^êêçï=oìäÉF=
=
=
Figure 72.
=
516. k-íÜ=lêÇÉê=aÉíÉêãáå~åí=
åå
åà
O
å
N
å
áå
áà
O
á
N
á
å
O
à
O
OO
ON
å
N
à
N
NO
NN
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
^
ÇÉí
K
K
K
K
K
K
K
K
K
K
K
K
K
K
K
K
K
K
K
K
= =
=
517. jáåçê=
qÜÉ=ãáåçê= áà
j =~ëëçÅá~íÉÇ=ïáíÜ=íÜÉ=ÉäÉãÉåí= áà
~ =çÑ=å-íÜ=çêÇÉê=
ã~íêáñ=^=áë=íÜÉ= ( )
N
å − -íÜ=çêÇÉê=ÇÉíÉêãáå~åí=ÇÉêáîÉÇ=Ñêçã=
íÜÉ=ã~íêáñ=^=Äó=ÇÉäÉíáçå=çÑ=áíë=á-íÜ=êçï=~åÇ=à-íÜ=ÅçäìãåK===
=
119. CHAPTER 5. MATRICES AND DETERMINANTS
109
518. `çÑ~Åíçê=
( ) áà
à
á
áà j
N
`
+
−
= =
=
519. i~éä~ÅÉ=bñé~åëáçå=çÑ=å-íÜ=lêÇÉê=aÉíÉêãáå~åí=
i~éä~ÅÉ=Éñé~åëáçå=Äó=ÉäÉãÉåíë=çÑ=íÜÉ=á-íÜ=êçï=
∑
=
=
å
N
à
áà
áà`
~
^
ÇÉí I= å
I
I
O
I
N
á K
= K=
i~éä~ÅÉ=Éñé~åëáçå=Äó=ÉäÉãÉåíë=çÑ=íÜÉ=à-íÜ=Åçäìãå=
∑
=
=
å
N
á
áà
áà`
~
^
ÇÉí I= å
I
I
O
I
N
à K
= K==
=
=
=
5.2 Properties of Determinants
=
520. qÜÉ==î~äìÉ==çÑ=~=ÇÉíÉêãáå~åí=êÉã~áåë==ìåÅÜ~åÖÉÇ=áÑ=êçïë=~êÉ=
ÅÜ~åÖÉÇ=íç=Åçäìãåë=~åÇ=Åçäìãåë=íç=êçïëK=
=
O
O
N
N
O
N
O
N
Ä
~
Ä
~
Ä
Ä
~
~
= ==
=
521. fÑ=íïç==êçïë==Eçê=íïç=ÅçäìãåëF=~êÉ==áåíÉêÅÜ~åÖÉÇI=íÜÉ=ëáÖå=çÑ=
íÜÉ=ÇÉíÉêãáå~åí=áë=ÅÜ~åÖÉÇK=
N
N
O
O
O
O
N
N
Ä
~
Ä
~
Ä
~
Ä
~
−
= =
=
522. fÑ=íïç=êçïë==Eçê=íïç=ÅçäìãåëF=~êÉ==áÇÉåíáÅ~äI=íÜÉ=î~äìÉ=çÑ=íÜÉ=
ÇÉíÉêãáå~åí=áë=òÉêçK=
M
~
~
~
~
O
O
N
N
= =
=
120. CHAPTER 5. MATRICES AND DETERMINANTS
110
523. fÑ==íÜÉ===ÉäÉãÉåíë==çÑ==~åó=êçï==Eçê=ÅçäìãåF=~êÉ=ãìäíáéäáÉÇ=Äó=====
~==Åçããçå==Ñ~ÅíçêI==íÜÉ==ÇÉíÉêãáå~åí==áë==ãìäíáéäáÉÇ==Äó==íÜ~í=
Ñ~ÅíçêK=
O
O
N
N
O
O
N
N
Ä
~
Ä
~
â
Ä
~
âÄ
â~
= =
=
524. fÑ==íÜÉ==ÉäÉãÉåíë==çÑ==~åó==êçï==Eçê==ÅçäìãåF=~êÉ=áåÅêÉ~ëÉÇ=Eçê=
ÇÉÅêÉ~ëÉÇFÄó=Éèì~ä=ãìäíáéäÉë=çÑ=íÜÉ=ÅçêêÉëéçåÇáåÖ=ÉäÉãÉåíë=
çÑ=~åó=çíÜÉê=êçï==Eçê=ÅçäìãåFI==íÜÉ=î~äìÉ=çÑ=íÜÉ=ÇÉíÉêãáå~åí=
áë=ìåÅÜ~åÖÉÇK=
O
O
N
N
O
O
O
N
N
N
Ä
~
Ä
~
Ä
âÄ
~
Ä
âÄ
~
=
+
+
=
=
=
=
5.3 Matrices
=
525. aÉÑáåáíáçå=
^å= å
ã× =ã~íêáñ=^=áë=~=êÉÅí~åÖìä~ê=~êê~ó=çÑ=ÉäÉãÉåíë=Eåìã-
ÄÉêë=çê=ÑìåÅíáçåëF=ïáíÜ=ã=êçïë=~åÇ=å=ÅçäìãåëK==
[ ]
=
=
ãå
O
ã
N
ã
å
O
OO
ON
å
N
NO
NN
áà
~
~
~
~
~
~
~
~
~
~
^
K
M
M
M
K
K
==
=
526. pèì~êÉ=ã~íêáñ=áë=~=ã~íêáñ=çÑ=çêÇÉê= å
å× K==
=
527. ^=ëèì~êÉ=ã~íêáñ==[ ]
áà
~ ==áë==ëóããÉíêáÅ==áÑ== àá
áà ~
~ = I==áKÉK==áí==áë=
ëóããÉíêáÅ=~Äçìí=íÜÉ=äÉ~ÇáåÖ=Çá~Öçå~äK==
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528. ^=ëèì~êÉ=ã~íêáñ=[ ]
áà
~ =áë=ëâÉï-ëóããÉíêáÅ=áÑ= àá
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= K==
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121. CHAPTER 5. MATRICES AND DETERMINANTS
111
529. aá~Öçå~ä=ã~íêáñ==áë==~=ëèì~êÉ==ã~íêáñ=ïáíÜ=~ää==ÉäÉãÉåíë==òÉêç=
ÉñÅÉéí=íÜçëÉ=çå=íÜÉ=äÉ~ÇáåÖ=Çá~Öçå~äK==
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531. ^=åìää=ã~íêáñ=áë=çåÉ=ïÜçëÉ=ÉäÉãÉåíë=~êÉ=~ää=òÉêçK=
=
=
=
5.4 Operations with Matrices
=
532. qïç=ã~íêáÅÉë=^=~åÇ=_=~êÉ=Éèì~ä=áÑI=~åÇ=çåäó=áÑI=íÜÉó=~êÉ=ÄçíÜ=
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[ ]
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122. CHAPTER 5. MATRICES AND DETERMINANTS
112
íÜÉå==
+
+
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+
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534. fÑ=â=áë=~=ëÅ~ä~êI=~åÇ= [ ]
áà
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^ = =áë=~=ã~íêáñI=íÜÉå=
[ ]
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535. jìäíáéäáÅ~íáçå=çÑ=qïç=j~íêáÅÉë=
qïç= ã~íêáÅÉë= Å~å= ÄÉ= ãìäíáéäáÉÇ= íçÖÉíÜÉê= çåäó= ïÜÉå= íÜÉ=
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N
å
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123. CHAPTER 5. MATRICES AND DETERMINANTS
113
íÜÉå==
=
=
ãâ
O
ã
N
ã
â
O
OO
ON
â
N
NO
NN
Å
Å
Ä
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Å
Å
Å
`
^_
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M
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I==
ïÜÉêÉ==
∑
=
λ
λ
λ
=
+
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à
á
åà
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à
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à
N
N
á
áà Ä
~
Ä
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Ä
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Ä
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Å K =
E ã
I
I
O
I
N
á K
= X â
I
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à K
= FK==
=
qÜìë=áÑ=
[ ]
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OP
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=
⋅
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536. qê~åëéçëÉ=çÑ=~=j~íêáñ=
fÑ=íÜÉ=êçïë=~åÇ=Åçäìãåë=çÑ=~=ã~íêáñ=~êÉ=áåíÉêÅÜ~åÖÉÇI=íÜÉå=
íÜÉ=åÉï=ã~íêáñ=áë=Å~ääÉÇ=íÜÉ=íê~åëéçëÉ=çÑ=íÜÉ=çêáÖáå~ä=ã~íêáñK===
fÑ=^=áë=íÜÉ=çêáÖáå~ä=ã~íêáñI=áíë=íê~åëéçëÉ=áë=ÇÉåçíÉÇ= q
^ =çê=
^
ú
K==
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537. qÜÉ=ã~íêáñ=^=áë=çêíÜçÖçå~ä=áÑ= f
^^q
= K==
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538. fÑ=íÜÉ=ã~íêáñ=éêçÇìÅí=^_=áë=ÇÉÑáåÉÇI=íÜÉå==
( ) q
q
q
^
_
^_ = K=
=
=
124. CHAPTER 5. MATRICES AND DETERMINANTS
114
539. ^Çàçáåí=çÑ=j~íêáñ=
fÑ=^=áë=~=ëèì~êÉ= å
å× ã~íêáñI=áíë=~ÇàçáåíI=ÇÉåçíÉÇ=Äó= ^
~Çà I=
áë=íÜÉ=íê~åëéçëÉ=çÑ=íÜÉ=ã~íêáñ=çÑ=ÅçÑ~Åíçêë= áà
` =çÑ=^W=
[ ]q
áà
`
^
~Çà = K==
=
540. qê~ÅÉ=çÑ=~=j~íêáñ=
fÑ=^=áë=~=ëèì~êÉ= å
å× ã~íêáñI=áíë=íê~ÅÉI=ÇÉåçíÉÇ=Äó= ^
íê I=áë=
ÇÉÑáåÉÇ=íç=ÄÉ==íÜÉ=ëìã=çÑ==íÜÉ=íÉêãë=çå=íÜÉ=äÉ~ÇáåÖ=Çá~Öçå~äW=
åå
OO
NN ~
~
~
^
íê +
+
+
= K K=
=
541. fåîÉêëÉ=çÑ=~=j~íêáñ=
fÑ=^=áë=~=ëèì~êÉ= å
å× ã~íêáñ=ïáíÜ=~=åçåëáåÖìä~ê=ÇÉíÉêãáå~åí=
^
ÇÉí I=íÜÉå=áíë=áåîÉêëÉ= N
^−
=áë=ÖáîÉå=Äó=
^
ÇÉí
^
~Çà
^ N
=
−
K=
=
542. fÑ=íÜÉ=ã~íêáñ=éêçÇìÅí=^_=áë=ÇÉÑáåÉÇI=íÜÉå==
( ) N
N
N
^
_
^_ −
−
−
= K=
=
543. fÑ==^==áë=~=ëèì~êÉ=== å
å× ==ã~íêáñI==íÜÉ==ÉáÖÉåîÉÅíçêë==u===ë~íáëÑó=
íÜÉ=Éèì~íáçå=
u
^u λ
= I==
ïÜáäÉ=íÜÉ=ÉáÖÉåî~äìÉë=λ =ë~íáëÑó=íÜÉ=ÅÜ~ê~ÅíÉêáëíáÅ=Éèì~íáçå=
M
f
^ =
λ
− K===
=
=
=
5.5 Systems of Linear Equations
=
=
s~êá~ÄäÉëW=ñI=óI=òI= N
ñ I= K
I
ñO =
oÉ~ä=åìãÄÉêëW= K
I
~
I
~
I
Ä
I
~
I
~
I
~ NO
NN
N
P
O
N =
125. CHAPTER 5. MATRICES AND DETERMINANTS
115
aÉíÉêãáå~åíëW=aI= ñ
a I= ó
a I= ò
a ==
j~íêáÅÉëW=^I=_I=u=
=
=
544.
=
+
=
+
O
O
O
N
N
N
Ç
ó
Ä
ñ
~
Ç
ó
Ä
ñ
~
I==
a
a
ñ ñ
= I=
a
a
ó
ó
= =E`ê~ãÉê∞ë=êìäÉFI==
ïÜÉêÉ==
N
O
O
N
O
O
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N
Ä
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Ä
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a −
=
= I==
N
O
O
N
O
O
N
N
ñ Ä
Ç
Ä
Ç
Ä
Ç
Ä
Ç
a −
=
= I==
N
O
O
N
O
O
N
N
ó Ç
~
Ç
~
Ç
~
Ç
~
a −
=
= K==
=
545. fÑ= M
a ≠ I=íÜÉå=íÜÉ=ëóëíÉã=Ü~ë=~=ëáåÖäÉ=ëçäìíáçåW==
a
a
ñ ñ
= I=
a
a
ó
ó
= K=
fÑ= M
a = =~åÇ= M
añ ≠ Eçê= M
aó ≠ FI=íÜÉå=íÜÉ=ëóëíÉã=Ü~ë==åç==
ëçäìíáçåK=
fÑ= M
a
a
a ó
ñ =
=
= I= íÜÉå= íÜÉ= ëóëíÉã= Ü~ë= = áåÑáåáíÉäó= = ã~åó==
ëçäìíáçåëK=
=
546.
=
+
+
=
+
+
=
+
+
P
P
P
P
O
O
O
O
N
N
N
N
Ç
ò
Å
ó
Ä
ñ
~
Ç
ò
Å
ó
Ä
ñ
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=
Ç
ò
Å
ó
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I==
a
a
ñ ñ
= I=
a
a
ó
ó
= I=
a
a
ò ò
= =E`ê~ãÉê∞ë=êìäÉFI==
=
126. CHAPTER 5. MATRICES AND DETERMINANTS
116
ïÜÉêÉ==
P
P
P
O
O
O
N
N
N
Å
Ä
~
Å
Ä
~
Å
Ä
~
a = I=
P
P
P
O
O
O
N
N
N
ñ
Å
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Å
Ä
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Å
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a = I=
P
P
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N
N
ó
Å
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~
Å
Ç
~
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a = I=
P
P
P
O
O
O
N
N
N
ò
Ç
Ä
~
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Ä
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Ä
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a = K==
=
547. fÑ= M
a ≠ I=íÜÉå=íÜÉ=ëóëíÉã=Ü~ë=~=ëáåÖäÉ=ëçäìíáçåW==
a
a
ñ ñ
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a
a
ó
ó
= I=
a
a
ò ò
= K=
fÑ= M
a = =~åÇ= M
añ ≠ Eçê= M
aó ≠ =çê= M
aò ≠ FI=íÜÉå=íÜÉ=ëóëíÉã=
Ü~ë=åç=ëçäìíáçåK=
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a
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ó
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548. j~íêáñ=cçêã=çÑ=~=póëíÉã=çÑ=å=iáåÉ~ê=bèì~íáçåë=áå=================
å=råâåçïåë=
qÜÉ=ëÉí=çÑ=äáåÉ~ê=Éèì~íáçåë==
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=
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M
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u
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⋅ I==
127. CHAPTER 5. MATRICES AND DETERMINANTS
117
ïÜÉêÉ==
=
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O
å
N
å
å
O
OO
ON
å
N
NO
NN
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N
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N
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M
K==
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549. pçäìíáçå=çÑ=~=pÉí=çÑ=iáåÉ~ê=bèì~íáçåë= å
å× =
_
^
u N
⋅
= −
I==
ïÜÉêÉ= N
^−
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=
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128. 118
Chapter 6
Vectors
=
=
=
=
sÉÅíçêëW=ì
r
I= î
r
I= ï
r
I= ê
r
I=
→
^_ I=£=
sÉÅíçê=äÉåÖíÜW= ì
r
I= î
r
I=£=
råáí=îÉÅíçêëW= á
r
I= à
r
I=â
r
=
kìää=îÉÅíçêW=M
r
=
`ççêÇáå~íÉë=çÑ=îÉÅíçê=ì
r
W= N
N
N w
I
v
I
u =
`ççêÇáå~íÉë=çÑ=îÉÅíçê= î
r
W= O
O
O w
I
v
I
u =
pÅ~ä~êëW=λ Iµ=
aáêÉÅíáçå=ÅçëáåÉëW= α
Åçë I= β
Åçë I= γ
Åçë =
^åÖäÉ=ÄÉíïÉÉå=íïç=îÉÅíçêëW=θ =
=
=
6.1 Vector Coordinates
=
550. råáí=sÉÅíçêë=
( )
M
I
M
I
N
á =
r
I=
( )
M
I
N
I
M
à =
r
I=
( )
N
I
M
I
M
â =
r
I=
N
â
à
á =
=
=
r
r
r
K=
=
551. ( ) ( ) ( )â
ò
ò
à
ó
ó
á
ñ
ñ
^_
ê M
N
M
N
M
N
r
r
r
r
−
+
−
+
−
=
=
→
=
=
129. CHAPTER 6. VECTORS
119
======= =
=
Figure 73.
=
552. ( ) ( ) ( )O
M
N
O
M
N
O
M
N ò
ò
ó
ó
ñ
ñ
^_
ê −
+
−
+
−
=
=
→
r
=
=
553. fÑ= ê
^_
r
=
→
I=íÜÉå= ê
_^
r
−
=
→
K=
=
=
=
Figure 74.
=
554. α
= Åçë
ê
u
r
I=
β
= Åçë
ê
v
r
I=
γ
= Åçë
ê
w
r
K=
130. CHAPTER 6. VECTORS
120
===== =
=
Figure 75.
=
555. fÑ= ( ) ( )
N
N
N
N w
I
v
I
u
ê
w
I
v
I
u
ê
r
r
= I=íÜÉå==
N
u
u = I= N
v
v = I= N
w
w = K==
==
=
6.2 Vector Addition
=
556. î
ì
ï
r
r
r
+
= =
=
== =
=
Figure 76.
131. CHAPTER 6. VECTORS
121
== =
=
Figure 77.
=
557. å
P
O
N ì
ì
ì
ì
ï
r
K
r
r
r
r
+
+
+
+
= =
=
== =
=
Figure 78.
=
558. `çããìí~íáîÉ=i~ï=
ì
î
î
ì
r
r
r
r
+
=
+ =
=
559. ^ëëçÅá~íáîÉ=i~ï=
( ) ( )
ï
î
ì
ï
î
ì
r
r
r
r
r
r
+
+
=
+
+ =
=
560. ( )
O
N
O
N
O
N w
w
I
v
v
I
u
u
î
ì +
+
+
=
+
r
r
=
=
=
=
=
=
=
132. CHAPTER 6. VECTORS
122
6.3 Vector Subtraction
=
561. î
ì
ï
r
r
r
−
= =áÑ= ì
ï
î
r
r
r
=
+ K=
=
=
=
Figure 79.
=
== =
=
Figure 80.
=
562. ( )
î
ì
î
ì
r
r
r
r
−
+
=
− =
=
563. ( )
M
I
M
I
M
M
ì
ì =
=
−
r
r
r
=
=
564. M
M =
r
=
=
565. ( )
O
N
O
N
O
N w
w
I
v
v
I
u
u
î
ì −
−
−
=
−
r
r
I==
=
=
=
6.4 Scaling Vectors
=
566. ì
ï
r
r
λ
= =
133. CHAPTER 6. VECTORS
123
=
=
Figure 81.
=
567. ì
ï
r
r
⋅
λ
= =
=
568. ( )
w
I
v
I
u
ì λ
λ
λ
=
λ
r
=
=
569. λ
=
λ ì
ì
r
r
=
=
570. ( ) ì
ì
ì
r
r
r
µ
+
λ
=
µ
+
λ =
=
571. ( ) ( ) ( )ì
ì
ì
r
r
r
λµ
=
λ
µ
=
µ
λ =
=
572. ( ) î
ì
î
ì
r
r
r
r
λ
+
λ
=
+
λ =
=
=
=
6.5 Scalar Product
=
573. pÅ~ä~ê=mêçÇìÅí=çÑ=sÉÅíçêë=ì
r
=~åÇ î
r
=
θ
⋅
⋅
=
⋅ Åçë
î
ì
î
ì
r
r
r
r
I==
ïÜÉêÉ=θ =áë=íÜÉ=~åÖäÉ=ÄÉíïÉÉå=îÉÅíçêë=ì
r
=~åÇ î
r
K====
=
134. CHAPTER 6. VECTORS
124
= =
=
Figure 82.
=
574. pÅ~ä~ê=mêçÇìÅí=áå=`ççêÇáå~íÉ=cçêã=
fÑ= ( )
N
N
N w
I
v
I
u
ì =
r
I= ( )
O
O
O w
I
v
I
u
î =
r
I=íÜÉå==
O
N
O
N
O
N w
w
v
v
u
u
î
ì +
+
=
⋅
r
r
K=
=
575. ^åÖäÉ=_ÉíïÉÉå=qïç=sÉÅíçêë==
fÑ= ( )
N
N
N w
I
v
I
u
ì =
r
I= ( )
O
O
O w
I
v
I
u
î =
r
I=íÜÉå==
O
O
O
O
O
O
O
N
O
N
O
N
O
N
O
N
O
N
w
v
u
w
v
u
w
w
v
v
u
u
Åçë
+
+
+
+
+
+
=
θ K=
=
576. `çããìí~íáîÉ=mêçéÉêíó=
ì
î
î
ì
r
r
r
r
⋅
=
⋅ =
=
577. ^ëëçÅá~íáîÉ=mêçéÉêíó=
( ) ( ) î
ì
î
ì
r
r
r
r
⋅
λµ
=
µ
⋅
λ =
=
578. aáëíêáÄìíáîÉ=mêçéÉêíó=
( ) ï
ì
î
ì
ï
î
ì
r
r
r
r
r
r
r
⋅
+
⋅
=
+
⋅ =
=
579. M
î
ì =
⋅
r
r
=áÑ=ì
r
I î
r
=~êÉ=çêíÜçÖçå~ä=E
O
π
=
θ FK=
=
580. M
î
ì >
⋅
r
r
=áÑ=
O
M
π
<
θ
< K=
=
135. CHAPTER 6. VECTORS
125
581. M
î
ì <
⋅
r
r
=áÑ= π
<
θ
<
π
O
K=
=
582. î
ì
î
ì
r
r
r
r
⋅
≤
⋅ =
=
583. î
ì
î
ì
r
r
r
r
⋅
=
⋅ =áÑ=ì
r
I î
r
=~êÉ=é~ê~ääÉä=E M
=
θ FK=
=
584. fÑ= ( )
N
N
N w
I
v
I
u
ì =
r
I=íÜÉå==
O
N
O
N
O
N
O
O
w
v
u
ì
ì
ì
ì +
+
=
=
=
⋅
r
r
r
r
K=
=
585. N
â
â
à
à
á
á =
⋅
=
⋅
=
⋅
r
r
r
r
r
r
=
=
586. M
á
â
â
à
à
á =
⋅
=
⋅
=
⋅
r
r
r
r
r
r
=
=
=
=
6.6 Vector Product
=
587. sÉÅíçê=mêçÇìÅí=çÑ=sÉÅíçêë=ì
r
=~åÇ î
r
=
ï
î
ì
r
r
r
=
× I=ïÜÉêÉ==
• θ
⋅
⋅
= ëáå
î
ì
ï
r
r
r
I=ïÜÉêÉ=
O
M
π
≤
θ
≤ X=
• ì
ï
r
r
⊥ = ~åÇ= î
ï
r
r
⊥ X=
• =sÉÅíçêë=ì
r
I= î
r
I= ï
r
=Ñçêã=~=êáÖÜí-Ü~åÇÉÇ=ëÅêÉïK=
=
136. CHAPTER 6. VECTORS
126
======= =
=
Figure 83.
=
588.
O
O
O
N
N
N
w
v
u
w
v
u
â
à
á
î
ì
ï
r
r
r
r
r
r
=
×
= =
=
589.
−
=
×
=
O
O
N
N
O
O
N
N
O
O
N
N
v
u
v
u
I
w
u
w
u
I
w
v
w
v
î
ì
ï
r
r
r
=
=
590. θ
⋅
⋅
=
×
= ëáå
î
ì
î
ì
p
r
r
r
r
=EcáÖKUPF=
=
591. ^åÖäÉ=_ÉíïÉÉå=qïç=sÉÅíçêë=EcáÖKUPF=
î
ì
î
ì
ëáå r
r
r
r
⋅
×
=
θ =
=
592. kçåÅçããìí~íáîÉ=mêçéÉêíó=
( )
ì
î
î
ì
r
r
r
r
×
−
=
× ==
=
593. ^ëëçÅá~íáîÉ=mêçéÉêíó=
( ) ( ) î
ì
î
ì
r
r
r
r
×
λµ
=
µ
×
λ =
=
=
137. CHAPTER 6. VECTORS
127
594. aáëíêáÄìíáîÉ=mêçéÉêíó=
( ) ï
ì
î
ì
ï
î
ì
r
r
r
r
r
r
r
×
+
×
=
+
× =
=
595. M
î
ì
r
r
r
=
× =áÑ=ì
r
=~åÇ= î
r
=~êÉ=é~ê~ääÉä=E M
=
θ FK=
=
596. M
â
â
à
à
á
á
r
r
r
r
r
r
r
=
×
=
×
=
× =
=
597. â
à
á
r
r
r
=
× I= á
â
à
r
r
r
=
× I= à
á
â
r
r
r
=
× =
=
=
=
6.7 Triple Product
=
598. pÅ~ä~ê=qêáéäÉ=mêçÇìÅí=
[ ] ( ) ( ) ( )
î
ì
ï
ì
ï
î
ï
î
ì
ï
î
ì
r
r
r
r
r
r
r
r
r
r
r
r
×
⋅
=
×
⋅
=
×
⋅
= =
=
599. [ ] [ ] [ ] [ ] [ ] [ ]
î
ï
ì
ì
î
ï
ï
ì
î
ì
ï
î
î
ì
ï
ï
î
ì
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
−
=
−
=
−
=
=
= =
=
600. ( ) [ ]
ï
î
ì
â
ï
î
ì
â
r
r
r
r
r
r
=
×
⋅ =
=
601. pÅ~ä~ê=qêáéäÉ=mêçÇìÅí=áå=`ççêÇáå~íÉ=cçêã=
( )
P
P
P
O
O
O
N
N
N
w
v
u
w
v
u
w
v
u
ï
î
ì =
×
⋅
r
r
r
I==
ïÜÉêÉ==
( )
N
N
N w
I
v
I
u
ì =
r
I= ( )
O
O
O w
I
v
I
u
î =
r
I= ( )
P
P
P w
I
v
I
u
ï =
r
K==
=
602. sçäìãÉ=çÑ=m~ê~ääÉäÉéáéÉÇ=
( )
ï
î
ì
s
r
r
r
×
⋅
= =
=
138. CHAPTER 6. VECTORS
128
============ =
=
Figure 84.
=
603. sçäìãÉ=çÑ=móê~ãáÇ=
( )
ï
î
ì
S
N
s
r
r
r
×
⋅
= =
=
=
=
Figure 85.
=
604. fÑ== ( ) M
ï
î
ì =
×
⋅
r
r
r
I=íÜÉå=íÜÉ=îÉÅíçêë==ì
r
I= î
r
I=~åÇ= ï
r
=~êÉ=äáåÉ~êäó=
ÇÉéÉåÇÉåí=I=ëç= î
ì
ï
r
r
r
µ
+
λ
= =Ñçê=ëçãÉ=ëÅ~ä~êë=λ =~åÇ=µK==
=
605. fÑ== ( ) M
ï
î
ì ≠
×
⋅
r
r
r
I=íÜÉå=íÜÉ=îÉÅíçêë==ì
r
I= î
r
I=~åÇ= ï
r
=~êÉ=äáåÉ~êäó=
áåÇÉéÉåÇÉåíK=
=
139. CHAPTER 6. VECTORS
129
606. sÉÅíçê=qêáéäÉ=mêçÇìÅí=
( ) ( ) ( )ï
î
ì
î
ï
ì
ï
î
ì
r
r
r
r
r
r
r
r
r
⋅
−
⋅
=
×
× ==
=
=
=
=
=
=
=
=
140. 130
Chapter 7
Analytic Geometry
=
=
=
=
7.1 One-Dimensional Coordinate System
=
mçáåí=ÅççêÇáå~íÉëW= M
ñ I= N
ñ I= O
ñ I= M
ó I= N
ó I= O
ó =
oÉ~ä=åìãÄÉêW=λ ==
aáëí~åÅÉ=ÄÉíïÉÉå=íïç=éçáåíëW=Ç=
=
=
607. aáëí~åÅÉ=_ÉíïÉÉå=qïç=mçáåíë=
O
N
N
O ñ
ñ
ñ
ñ
^_
Ç −
=
−
=
= =
=
=
=
Figure 86.
=
608. aáîáÇáåÖ=~=iáåÉ=pÉÖãÉåí=áå=íÜÉ=o~íáç=λ =
λ
+
λ
+
=
N
ñ
ñ
ñ O
N
M I=
`_
^`
=
λ I= N
−
≠
λ K=
=
======== =
=
Figure 87.
141. CHAPTER 7. ANALYTIC GEOMETRY
131
609. jáÇéçáåí=çÑ=~=iáåÉ=pÉÖãÉåí=
O
ñ
ñ
ñ O
N
M
+
= I= N
=
λ K=
=
=
=
7.2 Two-Dimensional Coordinate System
=
mçáåí=ÅççêÇáå~íÉëW= M
ñ I= N
ñ I= O
ñ I= M
ó I= N
ó I= O
ó =
mçä~ê=ÅççêÇáå~íÉëW= ϕ
I
ê =
oÉ~ä=åìãÄÉêW=λ ==
mçëáíáîÉ=êÉ~ä=åìãÄÉêëW=~I=ÄI=ÅI==
aáëí~åÅÉ=ÄÉíïÉÉå=íïç=éçáåíëW=Ç=
^êÉ~W=p=
=
=
610. aáëí~åÅÉ=_ÉíïÉÉå=qïç=mçáåíë=
( ) ( )O
N
O
O
N
O ó
ó
ñ
ñ
^_
Ç −
+
−
=
= =
=
=
=
Figure 88.
142. CHAPTER 7. ANALYTIC GEOMETRY
132
611. aáîáÇáåÖ=~=iáåÉ=pÉÖãÉåí=áå=íÜÉ=o~íáç=λ =
λ
+
λ
+
=
N
ñ
ñ
ñ O
N
M I=
λ
+
λ
+
=
N
ó
ó
ó O
N
M I==
`_
^`
=
λ I= N
−
≠
λ K=
=
======= =
=
Figure 89.
=
=
143. CHAPTER 7. ANALYTIC GEOMETRY
133
======= =
=
Figure 90.
=
612. jáÇéçáåí=çÑ=~=iáåÉ=pÉÖãÉåí=
O
ñ
ñ
ñ O
N
M
+
= I=
O
ó
ó
ó O
N
M
+
= I= N
=
λ K=
=
613. `ÉåíêçáÇ=EfåíÉêëÉÅíáçå=çÑ=jÉÇá~åëF=çÑ=~=qêá~åÖäÉ=
P
ñ
ñ
ñ
ñ P
O
N
M
+
+
= I=
P
ó
ó
ó
ó P
O
N
M
+
+
= I==
ïÜÉêÉ== ( )
N
N ó
I
ñ
^ I== ( )
O
O ó
I
ñ
_ I==~åÇ== ( )
P
P ó
I
ñ
` ==~êÉ=îÉêíáÅÉë=çÑ=
íÜÉ=íêá~åÖäÉ= ^_` K= =
=
144. CHAPTER 7. ANALYTIC GEOMETRY
134
========= =
=
Figure 91.
=
614. fåÅÉåíÉê=EfåíÉêëÉÅíáçå=çÑ=^åÖäÉ=_áëÉÅíçêëF=çÑ=~=qêá~åÖäÉ=
Å
Ä
~
Åñ
Äñ
~ñ
ñ P
O
N
M
+
+
+
+
= I=
Å
Ä
~
Åó
Äó
~ó
ó P
O
N
M
+
+
+
+
= I==
ïÜÉêÉ= _`
~ = I= `^
Ä = I= ^_
Å = K==
=
======== =
=
Figure 92.
145. CHAPTER 7. ANALYTIC GEOMETRY
135
615. `áêÅìãÅÉåíÉê=EfåíÉêëÉÅíáçå=çÑ=íÜÉ=páÇÉ=mÉêéÉåÇáÅìä~ê======================
_áëÉÅíçêëF=çÑ=~=qêá~åÖäÉ=
N
ó
ñ
N
ó
ñ
N
ó
ñ
O
N
ó
ó
ñ
N
ó
ó
ñ
N
ó
ó
ñ
ñ
P
P
O
O
N
N
P
O
P
O
P
O
O
O
O
O
N
O
N
O
N
M
+
+
+
= I=
N
ó
ñ
N
ó
ñ
N
ó
ñ
O
N
ó
ñ
ñ
N
ó
ñ
ñ
N
ó
ñ
ñ
ó
P
P
O
O
N
N
O
P
O
P
P
O
O
O
O
O
O
N
O
N
N
M
+
+
+
= =
=
======== =
==
Figure 93.
=
=
=
=
=
=
=
146. CHAPTER 7. ANALYTIC GEOMETRY
136
616. lêíÜçÅÉåíÉê=EfåíÉêëÉÅíáçå=çÑ=^äíáíìÇÉëF=çÑ=~=qêá~åÖäÉ=
N
ó
ñ
N
ó
ñ
N
ó
ñ
N
ó
ñ
ñ
ó
N
ó
ñ
ñ
ó
N
ó
ñ
ñ
ó
ñ
P
P
O
O
N
N
O
P
O
N
P
O
O
N
P
O
O
N
P
O
N
M
+
+
+
= I=
N
ó
ñ
N
ó
ñ
N
ó
ñ
N
ñ
ó
ó
ñ
N
ñ
ó
ó
ñ
N
ñ
ó
ó
ñ
ó
P
P
O
O
N
N
P
O
N
O
P
O
N
P
O
O
N
P
O
O
N
M
+
+
+
= =
=
====== =
=
Figure 94.
=
617. ^êÉ~=çÑ=~=qêá~åÖäÉ=
( ) ( )
N
P
N
P
N
O
N
O
P
P
O
O
N
N
ó
ó
ñ
ñ
ó
ó
ñ
ñ
O
N
N
ó
ñ
N
ó
ñ
N
ó
ñ
O
N
p
−
−
−
−
±
=
±
= =
=
=
=
147. CHAPTER 7. ANALYTIC GEOMETRY
137
618. ^êÉ~=çÑ=~=nì~Çêáä~íÉê~ä=
( ) ( )( ) ( )( )
[ +
+
−
+
+
−
±
= P
O
P
O
O
N
O
N ó
ó
ñ
ñ
ó
ó
ñ
ñ
O
N
p =
( )( ) ( )( )]
N
Q
N
Q
Q
P
Q
P ó
ó
ñ
ñ
ó
ó
ñ
ñ +
−
+
+
−
+ =
=
=== =
=
Figure 95.
=
kçíÉW=få=Ñçêãìä~ë=SNTI=SNU=ïÉ=ÅÜççëÉ=íÜÉ=ëáÖå=EHF=çê=E¥F=ëç=
íÜ~í=íç=ÖÉí=~=éçëáíáîÉ=~åëïÉê=Ñçê=~êÉ~K==
=
619. aáëí~åÅÉ=_ÉíïÉÉå=qïç=mçáåíë=áå=mçä~ê=`ççêÇáå~íÉë=
( )
N
O
O
N
O
O
O
N Åçë
ê
ê
O
ê
ê
^_
Ç ϕ
−
ϕ
−
+
=
= =
=
148. CHAPTER 7. ANALYTIC GEOMETRY
138
=
=
Figure 96.
=
620. `çåîÉêíáåÖ=oÉÅí~åÖìä~ê=`ççêÇáå~íÉë=íç=mçä~ê=`ççêÇáå~íÉë=
ϕ
= Åçë
ê
ñ I= ϕ
= ëáå
ê
ó K=
=
=
=
Figure 97.
=
621. `çåîÉêíáåÖ=mçä~ê=`ççêÇáå~íÉë=íç=oÉÅí~åÖìä~ê=`ççêÇáå~íÉë=
O
O
ó
ñ
ê +
= I=
ñ
ó
í~å =
ϕ K=
149. CHAPTER 7. ANALYTIC GEOMETRY
139
7.3 Straight Line in Plane
=
mçáåí=ÅççêÇáå~íÉëW=uI=vI=ñI= M
ñ I= N
ñ I== M
ó I= N
ó I= N
~ I= O
~ I=£==
oÉ~ä=åìãÄÉêëW=âI=~I=ÄI=éI=íI=^I=_I=`I= N
^ I= O
^ I=£=
^åÖäÉëW=α I=β =
^åÖäÉ=ÄÉíïÉÉå=íïç=äáåÉëW=ϕ =
kçêã~ä=îÉÅíçêW=å
r
=
mçëáíáçå=îÉÅíçêëW= ê
r
I=~
r
I= Ä
r
=
=
=
622. dÉåÉê~ä=bèì~íáçå=çÑ=~=píê~áÖÜí=iáåÉ=
M
`
_ó
^ñ =
+
+ =
=
623. kçêã~ä=sÉÅíçê=íç=~=píê~áÖÜí=iáåÉ=
qÜÉ=îÉÅíçê= ( )
_
I
^
å
r
=áë=åçêã~ä=íç=íÜÉ=äáåÉ= M
`
_ó
^ñ =
+
+ K=
=
=
=
Figure 98.
=
624. bñéäáÅáí=bèì~íáçå=çÑ=~=píê~áÖÜí=iáåÉ=EpäçéÉ-fåíÉêÅÉéí=cçêãF=
Ä
âñ
ó +
= K==
150. CHAPTER 7. ANALYTIC GEOMETRY
140
qÜÉ=Öê~ÇáÉåí=çÑ=íÜÉ=äáåÉ=áë= α
= í~å
â K=
=
=
=
Figure 99.
=
625. dê~ÇáÉåí=çÑ=~=iáåÉ==
N
O
N
O
ñ
ñ
ó
ó
í~å
â
−
−
=
α
= =
=
=
=
Figure 100.
151. CHAPTER 7. ANALYTIC GEOMETRY
141
626. bèì~íáçå=çÑ=~=iáåÉ=dáîÉå=~=mçáåí=~åÇ=íÜÉ=dê~ÇáÉåí=
( )
M
M ñ
ñ
â
ó
ó −
+
= I==
ïÜÉêÉ=â=áë=íÜÉ=Öê~ÇáÉåíI= ( )
M
M ó
I
ñ
m =áë=~=éçáåí=çå=íÜÉ=äáåÉK=
=
=
=
Figure 101.
=
627. bèì~íáçå=çÑ=~=iáåÉ=qÜ~í=m~ëëÉë=qÜêçìÖÜ=qïç=mçáåíë=
N
O
N
N
O
N
ñ
ñ
ñ
ñ
ó
ó
ó
ó
−
−
=
−
−
==
çê=
M
N
ó
ñ
N
ó
ñ
N
ó
ñ
O
O
N
N = K=
=
153. CHAPTER 7. ANALYTIC GEOMETRY
143
629. kçêã~ä=cçêã=
M
é
ëáå
ó
Åçë
ñ =
−
β
+
β =
=
=
=
Figure 104.
=
630. mçáåí=aáêÉÅíáçå=cçêã=
v
ó
ó
u
ñ
ñ N
N −
=
−
I==
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145
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146
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157. CHAPTER 7. ANALYTIC GEOMETRY
147
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148
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159. CHAPTER 7. ANALYTIC GEOMETRY
149
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160. CHAPTER 7. ANALYTIC GEOMETRY
150
640. bèì~íáçå=çÑ=~=`áêÅäÉ=`ÉåíÉêÉÇ=~í=íÜÉ=lêáÖáå=Epí~åÇ~êÇ=
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161. CHAPTER 7. ANALYTIC GEOMETRY
151
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162. CHAPTER 7. ANALYTIC GEOMETRY
152
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163. CHAPTER 7. ANALYTIC GEOMETRY
153
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154
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165. CHAPTER 7. ANALYTIC GEOMETRY
155
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156
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157
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158
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159
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160
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161
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