This presentation will help learners to grasp and understand trigonometry concepts such as angles, triangles. It encompasses basic fundamental topics of trigonometry.
This document provides an overview of trigonometry including definitions and classifications of angles, triangles, and trigonometric functions. It discusses measuring angles in degrees and radians, converting between the two units, and calculating arc length, angular velocity, and linear velocity using trigonometric relationships. Examples are provided to illustrate key concepts related to angle measure, coterminal angles, circular motion, and the relationships between linear and angular quantities.
PC_Q2_W1-2_Angles in a Unit Circle Presentation PPTRichieReyes12
This document covers measures of arcs in a unit circle. It discusses angle measure in degrees and radians, how to convert between the two units, and illustrates angles in standard position and coterminal angles. It also explains that in a unit circle, an arc with length 1 intercepts a central angle measuring 1 radian. The length of an arc and area of a sector are directly proportional to the radian measure of the intercepting central angle.
The document discusses the formulas for calculating the arc length and area of a sector of a circle, stating that the arc length is equal to the radius multiplied by the central angle and the area of a sector is equal to one-half the radius squared multiplied by the central angle. It provides examples of using these formulas to solve problems involving finding the arc length or area of a sector given the radius and central angle.
This document discusses trigonometric functions. It begins by defining trigonometric functions as generalizations of trigonometric ratios to any angle measure, in terms of radian measure. It defines the six trigonometric functions - sine, cosine, tangent, cotangent, secant, and cosecant - in terms of the x-coordinate and y-coordinate of a point on a unit circle. Key properties discussed include the periodic nature of the functions and their values for quadrantal and other common angles.
This document provides an overview of fundamentals of trigonometry including:
- There are two main types of trigonometry - plane and spherical trigonometry. Plane trigonometry deals with angles and triangles in a plane, while spherical trigonometry deals with triangles on a sphere.
- An angle is defined as the union of two rays with a common endpoint, and can be measured in degrees or radians. There are four quadrants used to classify angles in the Cartesian plane.
- The trigonometric ratios of sine, cosine, and tangent are defined based on the sides of a right triangle containing the angle of interest. These ratios are fundamental functions in trigonometry.
* A rope is stretched from the top of a 12m tree to the ground
* The rope is 20m long
* Let's call the distance from the tree to the end of the rope x
* Then we have a right triangle with:
* Hypotenuse (c) = Rope length = 20m
* One leg (a) = Height of tree = 12m
* Other leg (x) = Distance from tree to end of rope
* Using the Pythagorean theorem: a^2 + b^2 = c^2
* x^2 + 12^2 = 20^2
* x^2 + 144 = 400
* x^2 = 256
* x =
This document defines terms related to angles, circular motion, and problem solving using angles. It provides formulas for calculating arc length from central angle measures, linear speed from arc length and time, and angular speed from angle and time. Examples demonstrate using the formulas to find arc lengths, speeds, and angle measures. Exercises provide additional practice problems involving angles, arc lengths, speeds, and real-world applications like highway signs and latitude/longitude.
This document provides an overview of trigonometry including definitions and classifications of angles, triangles, and trigonometric functions. It discusses measuring angles in degrees and radians, converting between the two units, and calculating arc length, angular velocity, and linear velocity using trigonometric relationships. Examples are provided to illustrate key concepts related to angle measure, coterminal angles, circular motion, and the relationships between linear and angular quantities.
PC_Q2_W1-2_Angles in a Unit Circle Presentation PPTRichieReyes12
This document covers measures of arcs in a unit circle. It discusses angle measure in degrees and radians, how to convert between the two units, and illustrates angles in standard position and coterminal angles. It also explains that in a unit circle, an arc with length 1 intercepts a central angle measuring 1 radian. The length of an arc and area of a sector are directly proportional to the radian measure of the intercepting central angle.
The document discusses the formulas for calculating the arc length and area of a sector of a circle, stating that the arc length is equal to the radius multiplied by the central angle and the area of a sector is equal to one-half the radius squared multiplied by the central angle. It provides examples of using these formulas to solve problems involving finding the arc length or area of a sector given the radius and central angle.
This document discusses trigonometric functions. It begins by defining trigonometric functions as generalizations of trigonometric ratios to any angle measure, in terms of radian measure. It defines the six trigonometric functions - sine, cosine, tangent, cotangent, secant, and cosecant - in terms of the x-coordinate and y-coordinate of a point on a unit circle. Key properties discussed include the periodic nature of the functions and their values for quadrantal and other common angles.
This document provides an overview of fundamentals of trigonometry including:
- There are two main types of trigonometry - plane and spherical trigonometry. Plane trigonometry deals with angles and triangles in a plane, while spherical trigonometry deals with triangles on a sphere.
- An angle is defined as the union of two rays with a common endpoint, and can be measured in degrees or radians. There are four quadrants used to classify angles in the Cartesian plane.
- The trigonometric ratios of sine, cosine, and tangent are defined based on the sides of a right triangle containing the angle of interest. These ratios are fundamental functions in trigonometry.
* A rope is stretched from the top of a 12m tree to the ground
* The rope is 20m long
* Let's call the distance from the tree to the end of the rope x
* Then we have a right triangle with:
* Hypotenuse (c) = Rope length = 20m
* One leg (a) = Height of tree = 12m
* Other leg (x) = Distance from tree to end of rope
* Using the Pythagorean theorem: a^2 + b^2 = c^2
* x^2 + 12^2 = 20^2
* x^2 + 144 = 400
* x^2 = 256
* x =
This document defines terms related to angles, circular motion, and problem solving using angles. It provides formulas for calculating arc length from central angle measures, linear speed from arc length and time, and angular speed from angle and time. Examples demonstrate using the formulas to find arc lengths, speeds, and angle measures. Exercises provide additional practice problems involving angles, arc lengths, speeds, and real-world applications like highway signs and latitude/longitude.
This section discusses angles and arcs. It defines angles, their measurement in degrees and radians, and how to convert between the two units. It also defines arc length and how to calculate it given the radius and measure of a central angle in radians. Finally, it discusses the relationships between linear speed, angular speed, radius, and time for objects moving in circular motion.
There are two main systems for measuring angles: the degree system and radian system. The degree system divides a full rotation into 360 equal degrees, while the radian system defines an angle as the arc length cut out on a unit circle. There are also two important types of right triangles used in trigonometry: the 45-45-90 triangle where the two legs have length a and the hypotenuse has length a√2, and the 30-60-90 triangle where one leg has length a, the other has length a/2, and the hypotenuse has length 2a.
This document provides information on 10 circle theorems including: angles in a semicircle are right angles; opposite angles in a cyclic quadrilateral add up to 180 degrees; equal chords subtend equal angles; and tangents from a point are equal in length. Examples are worked through demonstrating each theorem. Real-life applications are discussed such as using circle theorems and Pythagoras' theorem to calculate distances on Earth and how circle geometry has remained important in theories of atoms and the universe.
There are two systems for measuring angles: the degree system and the radian system. The degree system divides a full circle into 360 equal angles of 1 degree each. The radian system defines an angle as the arc length cut out by the angle on a unit circle of radius 1, where a full circle corresponds to 2π radians. While the degree system is commonly used, the radian system is preferred in mathematics due to its relationship to circle geometry formulas involving arc lengths and wedge areas.
This document discusses angle measure and special triangles. It defines angle, complementary angles, supplementary angles and coterminal angles. It then discusses 45-45-90 triangles and 30-60-90 triangles, stating that in 45-45-90 triangles the hypotenuse is twice the length of the legs and in 30-60-90 triangles the hypotenuse is twice the shorter leg and the longer leg is three times the shorter leg. It includes examples calculating side lengths of triangles.
This document provides information about plane and spherical trigonometry, including defining trigonometry, measuring angles in different units, finding arc length, and relating angular and linear velocity. It discusses specific objectives, defining an angle and different angle measurements. Examples are provided for converting between angle units, finding coterminal angles, and calculating arc length. Formulas are given for relating arc length to central angle measure and sector area to central angle measure. Angular and linear velocity are also defined.
This document provides information on trigonometric functions of right triangles. It defines the sine, cosine, and tangent functions as ratios of sides of a right triangle. It also introduces cosecant, secant, and cotangent as reciprocals of the primary trig functions. Several examples are given to calculate unknown side lengths or angles using trig functions. The document then covers trigonometric identities, angle sum and difference formulas, double and half angle formulas, and techniques for reducing trig powers and converting between sum and product formulas. It concludes with information on measuring angles in different units and introduces the study of oblique triangles, including the Law of Sines and Law of Cosines.
This module introduces the unit circle and trigonometric functions. It defines a unit circle as a circle with radius of 1 unit and discusses dividing the unit circle into congruent arcs. The module then covers converting between degrees and radians, defining angles intercepting arcs, and visualizing rotations along the unit circle. It concludes by discussing angles in standard position, quadrantal angles, and coterminal angles. Students are expected to learn key concepts like the unit circle, converting measures, and relating angles to arclengths and rotations.
The document discusses angles and their measurement in degrees and radians. It defines standard position of an angle, rotation of angles, coterminal angles, reference angles, and the relationship between degrees and radians. Examples are provided for finding measures of angles in standard position, coterminal angles, reference angles, and converting between degrees and radians. Key concepts covered include a full rotation being 360 degrees or 2π radians, and that coterminal angles have the same terminal side.
This document discusses angles and trigonometry concepts including:
- Angles in standard position, which have their initial side on the positive x-axis and can have their terminal side in any of the four quadrants.
- Coterminal angles, which share the same terminal side but have different measures, obtained by adding or subtracting multiples of 360°.
- Converting between degrees and radians, where one radian is the central angle subtended by an arc equal in length to the radius, and there are 2π radians in a full circle.
This document provides an overview of the Math 102 Trigonometry course. It will teach basic trigonometric concepts like evaluating the six trig functions, fundamental identities, and using right triangles. Students will learn to sketch trig function curves and use trigonometry to approximate the radius of Earth. Trigonometry deals with triangle measurements and angle units including degrees, radians, revolutions, and degrees-minutes-seconds. Conversions between these angle units are demonstrated along with calculating arc lengths on circles.
1) A circle is defined as the set of all points in a plane that are equidistant from a fixed point called the center.
2) Key terms related to circles include radius, diameter, chord, arc, sector, and segment. The radius connects the center to any point on the circle, while the diameter connects two points on the circle and passes through the center.
3) The area of a circle is calculated as πr^2, where r is the radius. The area of a sector of a circle is calculated based on the central angle in radians that the sector spans.
The document provides information about the basics of using a theodolite for angle measurements in surveying. It defines key terms like angle, vertex, and degrees. It describes the main components of a theodolite including the telescope, horizontal and vertical axes, plate bubbles, and screws. It explains how to perform temporary adjustments and measure both horizontal and vertical angles using methods like ordinary, repetition, and reiteration. Precise angle measurements are important for surveying applications like setting grades, ranging curves, and tachometric surveys.
Trigonometric ratios and functions deal with relationships in triangles, especially right triangles. They can be used both in geometry to solve triangles and analytically by treating trig functions as functions in equations. Trigonometry has been used for over 3000 years, originally to determine lengths and angles in triangles and more recently by treating trig functions as functions in equations. Angles are measured in degrees, minutes, and seconds, and can also be measured in radians where one radian is the central angle that intercepts the same arc length as the radius. Trig functions like sine, cosine, and tangent can be evaluated for any angle using the unit circle or by finding the reference angle in the first quadrant. Trig functions can be graphed by identifying
This document defines and describes different types of angles:
- Acute angles are less than 90 degrees. Obtuse angles are greater than 90 degrees but less than 180 degrees. Right angles are 90 degrees. Straight angles are 180 degrees. Reflex angles are greater than 180 degrees but less than 360 degrees.
- Angles can be calculated based on their relationship to other angles, such as angles around a point adding up to 360 degrees and angles on a straight line adding up to 180 degrees. Vertically opposite angles are always equal.
- When parallel lines are intersected by a transversal, the corresponding angles, alternate interior angles, alternate exterior angles, and interior angles on the same side of the transversal are
This document discusses triangles and their properties. It defines a triangle as a three-sided polygon with three angles and vertices. It describes the three main types of triangles based on side lengths (equilateral, isosceles, scalene) and angles (acute, right, obtuse). Heron's formula for calculating the area of a triangle given the side lengths is presented. Key properties of triangles like the angle sum property, exterior angle property, and congruency criteria (SSS, SAS, ASA) are outlined. Important triangle centers such as the incenter, circumcenter, centroid, and orthocenter are defined.
Areas related to Circles - class 10 maths Amit Choube
This a ppt which is based on chapter circles of class 10 maths it is a very good ppt which will definitely enhance your knowledge . it will also clear all concepts and doubts about this chapter and its topics
This document defines angles and angle measure in geometry and trigonometry. It explains that an angle is formed by two rays with a common endpoint, and can be measured in degrees from 0 to 360 degrees. The document discusses angle terminology like initial side, terminal side, standard position, coterminal angles, quadrantal angles, and locating angles by quadrant. It provides examples of finding coterminal angles and sketching angles in standard position. Exercises at the end have the reader practice finding coterminal angles, sketching angles, and determining angle locations.
This document provides definitions and explanations of terms related to horizontal curves. It discusses the following:
- Horizontal curves are used to connect two straight lines when there is a change in direction of a road or railway alignment. Circular curves are the most common type of horizontal curve.
- Key terms defined include degree of curve, radius, relationship between radius and degree, superelevation, and centrifugal ratio.
- Different types of horizontal curves are described, including simple circular, compound, reverse, and transition curves.
- Notation used in circular curves is explained, such as tangent points and lengths, deflection angle, and radius.
- Properties of simple circular curves are outlined, including
This section discusses angles and arcs. It defines angles, their measurement in degrees and radians, and how to convert between the two units. It also defines arc length and how to calculate it given the radius and measure of a central angle in radians. Finally, it discusses the relationships between linear speed, angular speed, radius, and time for objects moving in circular motion.
There are two main systems for measuring angles: the degree system and radian system. The degree system divides a full rotation into 360 equal degrees, while the radian system defines an angle as the arc length cut out on a unit circle. There are also two important types of right triangles used in trigonometry: the 45-45-90 triangle where the two legs have length a and the hypotenuse has length a√2, and the 30-60-90 triangle where one leg has length a, the other has length a/2, and the hypotenuse has length 2a.
This document provides information on 10 circle theorems including: angles in a semicircle are right angles; opposite angles in a cyclic quadrilateral add up to 180 degrees; equal chords subtend equal angles; and tangents from a point are equal in length. Examples are worked through demonstrating each theorem. Real-life applications are discussed such as using circle theorems and Pythagoras' theorem to calculate distances on Earth and how circle geometry has remained important in theories of atoms and the universe.
There are two systems for measuring angles: the degree system and the radian system. The degree system divides a full circle into 360 equal angles of 1 degree each. The radian system defines an angle as the arc length cut out by the angle on a unit circle of radius 1, where a full circle corresponds to 2π radians. While the degree system is commonly used, the radian system is preferred in mathematics due to its relationship to circle geometry formulas involving arc lengths and wedge areas.
This document discusses angle measure and special triangles. It defines angle, complementary angles, supplementary angles and coterminal angles. It then discusses 45-45-90 triangles and 30-60-90 triangles, stating that in 45-45-90 triangles the hypotenuse is twice the length of the legs and in 30-60-90 triangles the hypotenuse is twice the shorter leg and the longer leg is three times the shorter leg. It includes examples calculating side lengths of triangles.
This document provides information about plane and spherical trigonometry, including defining trigonometry, measuring angles in different units, finding arc length, and relating angular and linear velocity. It discusses specific objectives, defining an angle and different angle measurements. Examples are provided for converting between angle units, finding coterminal angles, and calculating arc length. Formulas are given for relating arc length to central angle measure and sector area to central angle measure. Angular and linear velocity are also defined.
This document provides information on trigonometric functions of right triangles. It defines the sine, cosine, and tangent functions as ratios of sides of a right triangle. It also introduces cosecant, secant, and cotangent as reciprocals of the primary trig functions. Several examples are given to calculate unknown side lengths or angles using trig functions. The document then covers trigonometric identities, angle sum and difference formulas, double and half angle formulas, and techniques for reducing trig powers and converting between sum and product formulas. It concludes with information on measuring angles in different units and introduces the study of oblique triangles, including the Law of Sines and Law of Cosines.
This module introduces the unit circle and trigonometric functions. It defines a unit circle as a circle with radius of 1 unit and discusses dividing the unit circle into congruent arcs. The module then covers converting between degrees and radians, defining angles intercepting arcs, and visualizing rotations along the unit circle. It concludes by discussing angles in standard position, quadrantal angles, and coterminal angles. Students are expected to learn key concepts like the unit circle, converting measures, and relating angles to arclengths and rotations.
The document discusses angles and their measurement in degrees and radians. It defines standard position of an angle, rotation of angles, coterminal angles, reference angles, and the relationship between degrees and radians. Examples are provided for finding measures of angles in standard position, coterminal angles, reference angles, and converting between degrees and radians. Key concepts covered include a full rotation being 360 degrees or 2π radians, and that coterminal angles have the same terminal side.
This document discusses angles and trigonometry concepts including:
- Angles in standard position, which have their initial side on the positive x-axis and can have their terminal side in any of the four quadrants.
- Coterminal angles, which share the same terminal side but have different measures, obtained by adding or subtracting multiples of 360°.
- Converting between degrees and radians, where one radian is the central angle subtended by an arc equal in length to the radius, and there are 2π radians in a full circle.
This document provides an overview of the Math 102 Trigonometry course. It will teach basic trigonometric concepts like evaluating the six trig functions, fundamental identities, and using right triangles. Students will learn to sketch trig function curves and use trigonometry to approximate the radius of Earth. Trigonometry deals with triangle measurements and angle units including degrees, radians, revolutions, and degrees-minutes-seconds. Conversions between these angle units are demonstrated along with calculating arc lengths on circles.
1) A circle is defined as the set of all points in a plane that are equidistant from a fixed point called the center.
2) Key terms related to circles include radius, diameter, chord, arc, sector, and segment. The radius connects the center to any point on the circle, while the diameter connects two points on the circle and passes through the center.
3) The area of a circle is calculated as πr^2, where r is the radius. The area of a sector of a circle is calculated based on the central angle in radians that the sector spans.
The document provides information about the basics of using a theodolite for angle measurements in surveying. It defines key terms like angle, vertex, and degrees. It describes the main components of a theodolite including the telescope, horizontal and vertical axes, plate bubbles, and screws. It explains how to perform temporary adjustments and measure both horizontal and vertical angles using methods like ordinary, repetition, and reiteration. Precise angle measurements are important for surveying applications like setting grades, ranging curves, and tachometric surveys.
Trigonometric ratios and functions deal with relationships in triangles, especially right triangles. They can be used both in geometry to solve triangles and analytically by treating trig functions as functions in equations. Trigonometry has been used for over 3000 years, originally to determine lengths and angles in triangles and more recently by treating trig functions as functions in equations. Angles are measured in degrees, minutes, and seconds, and can also be measured in radians where one radian is the central angle that intercepts the same arc length as the radius. Trig functions like sine, cosine, and tangent can be evaluated for any angle using the unit circle or by finding the reference angle in the first quadrant. Trig functions can be graphed by identifying
This document defines and describes different types of angles:
- Acute angles are less than 90 degrees. Obtuse angles are greater than 90 degrees but less than 180 degrees. Right angles are 90 degrees. Straight angles are 180 degrees. Reflex angles are greater than 180 degrees but less than 360 degrees.
- Angles can be calculated based on their relationship to other angles, such as angles around a point adding up to 360 degrees and angles on a straight line adding up to 180 degrees. Vertically opposite angles are always equal.
- When parallel lines are intersected by a transversal, the corresponding angles, alternate interior angles, alternate exterior angles, and interior angles on the same side of the transversal are
This document discusses triangles and their properties. It defines a triangle as a three-sided polygon with three angles and vertices. It describes the three main types of triangles based on side lengths (equilateral, isosceles, scalene) and angles (acute, right, obtuse). Heron's formula for calculating the area of a triangle given the side lengths is presented. Key properties of triangles like the angle sum property, exterior angle property, and congruency criteria (SSS, SAS, ASA) are outlined. Important triangle centers such as the incenter, circumcenter, centroid, and orthocenter are defined.
Areas related to Circles - class 10 maths Amit Choube
This a ppt which is based on chapter circles of class 10 maths it is a very good ppt which will definitely enhance your knowledge . it will also clear all concepts and doubts about this chapter and its topics
This document defines angles and angle measure in geometry and trigonometry. It explains that an angle is formed by two rays with a common endpoint, and can be measured in degrees from 0 to 360 degrees. The document discusses angle terminology like initial side, terminal side, standard position, coterminal angles, quadrantal angles, and locating angles by quadrant. It provides examples of finding coterminal angles and sketching angles in standard position. Exercises at the end have the reader practice finding coterminal angles, sketching angles, and determining angle locations.
This document provides definitions and explanations of terms related to horizontal curves. It discusses the following:
- Horizontal curves are used to connect two straight lines when there is a change in direction of a road or railway alignment. Circular curves are the most common type of horizontal curve.
- Key terms defined include degree of curve, radius, relationship between radius and degree, superelevation, and centrifugal ratio.
- Different types of horizontal curves are described, including simple circular, compound, reverse, and transition curves.
- Notation used in circular curves is explained, such as tangent points and lengths, deflection angle, and radius.
- Properties of simple circular curves are outlined, including
Similar to Angles, Triangles of Trigonometry. Pre - Calculus (20)
This slide is special for master students (MIBS & MIFB) in UUM. Also useful for readers who are interested in the topic of contemporary Islamic banking.
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Exploiting Artificial Intelligence for Empowering Researchers and Faculty, In...Dr. Vinod Kumar Kanvaria
Exploiting Artificial Intelligence for Empowering Researchers and Faculty,
International FDP on Fundamentals of Research in Social Sciences
at Integral University, Lucknow, 06.06.2024
By Dr. Vinod Kumar Kanvaria
Chapter wise All Notes of First year Basic Civil Engineering.pptxDenish Jangid
Chapter wise All Notes of First year Basic Civil Engineering
Syllabus
Chapter-1
Introduction to objective, scope and outcome the subject
Chapter 2
Introduction: Scope and Specialization of Civil Engineering, Role of civil Engineer in Society, Impact of infrastructural development on economy of country.
Chapter 3
Surveying: Object Principles & Types of Surveying; Site Plans, Plans & Maps; Scales & Unit of different Measurements.
Linear Measurements: Instruments used. Linear Measurement by Tape, Ranging out Survey Lines and overcoming Obstructions; Measurements on sloping ground; Tape corrections, conventional symbols. Angular Measurements: Instruments used; Introduction to Compass Surveying, Bearings and Longitude & Latitude of a Line, Introduction to total station.
Levelling: Instrument used Object of levelling, Methods of levelling in brief, and Contour maps.
Chapter 4
Buildings: Selection of site for Buildings, Layout of Building Plan, Types of buildings, Plinth area, carpet area, floor space index, Introduction to building byelaws, concept of sun light & ventilation. Components of Buildings & their functions, Basic concept of R.C.C., Introduction to types of foundation
Chapter 5
Transportation: Introduction to Transportation Engineering; Traffic and Road Safety: Types and Characteristics of Various Modes of Transportation; Various Road Traffic Signs, Causes of Accidents and Road Safety Measures.
Chapter 6
Environmental Engineering: Environmental Pollution, Environmental Acts and Regulations, Functional Concepts of Ecology, Basics of Species, Biodiversity, Ecosystem, Hydrological Cycle; Chemical Cycles: Carbon, Nitrogen & Phosphorus; Energy Flow in Ecosystems.
Water Pollution: Water Quality standards, Introduction to Treatment & Disposal of Waste Water. Reuse and Saving of Water, Rain Water Harvesting. Solid Waste Management: Classification of Solid Waste, Collection, Transportation and Disposal of Solid. Recycling of Solid Waste: Energy Recovery, Sanitary Landfill, On-Site Sanitation. Air & Noise Pollution: Primary and Secondary air pollutants, Harmful effects of Air Pollution, Control of Air Pollution. . Noise Pollution Harmful Effects of noise pollution, control of noise pollution, Global warming & Climate Change, Ozone depletion, Greenhouse effect
Text Books:
1. Palancharmy, Basic Civil Engineering, McGraw Hill publishers.
2. Satheesh Gopi, Basic Civil Engineering, Pearson Publishers.
3. Ketki Rangwala Dalal, Essentials of Civil Engineering, Charotar Publishing House.
4. BCP, Surveying volume 1
ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...PECB
Denis is a dynamic and results-driven Chief Information Officer (CIO) with a distinguished career spanning information systems analysis and technical project management. With a proven track record of spearheading the design and delivery of cutting-edge Information Management solutions, he has consistently elevated business operations, streamlined reporting functions, and maximized process efficiency.
Certified as an ISO/IEC 27001: Information Security Management Systems (ISMS) Lead Implementer, Data Protection Officer, and Cyber Risks Analyst, Denis brings a heightened focus on data security, privacy, and cyber resilience to every endeavor.
His expertise extends across a diverse spectrum of reporting, database, and web development applications, underpinned by an exceptional grasp of data storage and virtualization technologies. His proficiency in application testing, database administration, and data cleansing ensures seamless execution of complex projects.
What sets Denis apart is his comprehensive understanding of Business and Systems Analysis technologies, honed through involvement in all phases of the Software Development Lifecycle (SDLC). From meticulous requirements gathering to precise analysis, innovative design, rigorous development, thorough testing, and successful implementation, he has consistently delivered exceptional results.
Throughout his career, he has taken on multifaceted roles, from leading technical project management teams to owning solutions that drive operational excellence. His conscientious and proactive approach is unwavering, whether he is working independently or collaboratively within a team. His ability to connect with colleagues on a personal level underscores his commitment to fostering a harmonious and productive workplace environment.
Date: May 29, 2024
Tags: Information Security, ISO/IEC 27001, ISO/IEC 42001, Artificial Intelligence, GDPR
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Main Java[All of the Base Concepts}.docxadhitya5119
This is part 1 of my Java Learning Journey. This Contains Custom methods, classes, constructors, packages, multithreading , try- catch block, finally block and more.
বাংলাদেশের অর্থনৈতিক সমীক্ষা ২০২৪ [Bangladesh Economic Review 2024 Bangla.pdf] কম্পিউটার , ট্যাব ও স্মার্ট ফোন ভার্সন সহ সম্পূর্ণ বাংলা ই-বুক বা pdf বই " সুচিপত্র ...বুকমার্ক মেনু 🔖 ও হাইপার লিংক মেনু 📝👆 যুক্ত ..
আমাদের সবার জন্য খুব খুব গুরুত্বপূর্ণ একটি বই ..বিসিএস, ব্যাংক, ইউনিভার্সিটি ভর্তি ও যে কোন প্রতিযোগিতা মূলক পরীক্ষার জন্য এর খুব ইম্পরট্যান্ট একটি বিষয় ...তাছাড়া বাংলাদেশের সাম্প্রতিক যে কোন ডাটা বা তথ্য এই বইতে পাবেন ...
তাই একজন নাগরিক হিসাবে এই তথ্য গুলো আপনার জানা প্রয়োজন ...।
বিসিএস ও ব্যাংক এর লিখিত পরীক্ষা ...+এছাড়া মাধ্যমিক ও উচ্চমাধ্যমিকের স্টুডেন্টদের জন্য অনেক কাজে আসবে ...
Reimagining Your Library Space: How to Increase the Vibes in Your Library No ...Diana Rendina
Librarians are leading the way in creating future-ready citizens – now we need to update our spaces to match. In this session, attendees will get inspiration for transforming their library spaces. You’ll learn how to survey students and patrons, create a focus group, and use design thinking to brainstorm ideas for your space. We’ll discuss budget friendly ways to change your space as well as how to find funding. No matter where you’re at, you’ll find ideas for reimagining your space in this session.
This presentation includes basic of PCOS their pathology and treatment and also Ayurveda correlation of PCOS and Ayurvedic line of treatment mentioned in classics.
A workshop hosted by the South African Journal of Science aimed at postgraduate students and early career researchers with little or no experience in writing and publishing journal articles.
2. TRIGONOMETRY
• Derived from the Greek words “trigonon” which means
triangle and “metron” which means to measure.
• Branch of mathematics which deals with measurement of
triangles (i.e., their sides and angles), or more specifically,
with the indirect measurement of line segments and angles.
3. TRIANGLES
Definition: A triangle is a polygon with three sides and three
interior angles. The sum of the interior angles of a
triangle is 180°.
Classification of triangles according to angles:
• Oblique triangle – a triangle with no right angle
- Acute triangle
- Obtuse triangle
• Right triangle – a triangle with a right angle
• Equiangular triangle – a triangle with equal angles
4. TRIANGLES
Classification of triangles according to sides:
• Scalene Triangle - a triangle with no two sides equal.
• Isosceles Triangle - a triangle with two sides equal.
• Equilateral triangle – a triangle with three sides equal.
5. CLASSIFICATION OF ANGLES
• Zero angle – an angle of 0°.
• Acute angle – an angle between 0° and 90°.
• Right angle – an angle of 90°
• Obtuse angle – an angle between 90° and 180°
• Straight angle –an angle of 180°
• Reflex angle – an angle between 180° and 360°
• Circular angle – an angle of 360°
• Complex angle – an angle more than 360°
6. Lesson 1: ANGLE MEASURE
Math 12
Plane and Spherical Trigonometry
7. OBJECTIVES
At the end of the lesson the students are expected to:
• Measure angles in degrees and radians
• Define angles in standard position
• Convert degree measure to radian measure and vice versa
• Find the measures of coterminal angles
• Calculate the length of an arc along a circle.
• Solve problems involving arc length, angular velocity and
linear velocity
8. ANGLE
• An angle is formed by rotating a ray about its vertex from the
initial side to the terminal side.
• An angle is said to be in standard position if its initial side is
along the positive x-axis and its vertex is at the origin.
• Rotation in counterclockwise direction corresponds to a
positive angle.
• Rotation in clockwise direction corresponds to a negative
angle.
9. ANGLE MEASURE
The measure of an angle is the amount of rotation about the
vertex from the initial side to the terminal side.
Units of Measurement:
1. Degree
• denoted by °
• 1/360 of a complete rotation. One complete
counterclockwise rotation measures 360° , and one
complete clockwise rotation measures -360°.
2. Radian
• denoted by rad.
• measure of the central angle that is subtended by an arc
whose length is equal to the radius of the circle.
10. Definition: If a central angle 𝜃 in a circle with radius r
intercepts an arc on the circle of length s, then
𝜃 𝑖𝑛 𝑟𝑎𝑑𝑖𝑎𝑛𝑠 =
𝑠
𝑟
𝜃𝑓𝑢𝑙𝑙 𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛 ≈ 2𝜋 ≈ 360°
𝜋 ≈ 180°
11. CONVERTING BETWEEN DEGREES and
RADIANS
• To convert degrees to radians, multiply the degree measure
by
𝜋
180°
.
𝜃𝑟 = 𝜃𝑑
𝜋
180°
• To convert radians to degrees, multiply the radian measure by
180°
𝜋
.
𝜃𝑑 = 𝜃𝑟
180°
𝜋
12. Examples:
1. Find the degree measure of the angle for each rotation and
sketch each angle in standard position.
a)
1
2
rotation counterclockwise
b)
2
3
rotation clockwise
c)
5
9
rotation clockwise
d)
7
36
rotation counterclockwise
13. 2. Express each angle measure in radians. Give answers in
terms of 𝜋.
a) 60° c) -330°
b) 315° d) 780°
3. Express each angle measure in degrees.
a)
3𝜋
4
c) -
7𝜋
42
b)
11𝜋
9
d) 9𝜋
14. COTERMINAL ANGLES
Definition: Two angles in standard position with the same
terminal side are called coterminal angles.
Examples:
1. State in which quadrant the angles with the given measure in
standard position would be. Sketch each angle.
a) 145° c) -540°
b) 620° d) 1085°
15. COTERMINAL ANGLES
2. Determine the angle of the smallest possible positive
measure that is coterminal with each of the given angles.
a) 405° c) 960°
b) -135° d) 1350°
16. LENGTH OF A CIRCULAR ARC
Definition: If a central angle 𝜃 in a circle with radius r intercepts
an arc on the circle of length s, then the arc length s
is given by
𝑠 = 𝑟𝜃 𝜃 is in radians
r
S
17. LENGTH OF A CIRCULAR ARC
Examples:
1. Find the length of the arc intercepted by a central angle of
14° in a circle of radius of 15 cm.
2. The famous clock tower in London has a minute hand that is
14 feet long. How far does the tip of the minute hand of Big
Ben travel in 35 minutes?
3. The London Eye has 32 capsules and a diameter of 400 feet.
What is the distance you will have traveled once you reach
the highest point for the first time?
18. LINEAR SPEED
Definition: If a point P moves along the circumference of a circle
at a constant speed, then the linear speed v is given
by
𝑣 =
𝑠
𝑡
where s is the arc length and
t is the time.
19. ANGULAR SPEED
Definition: If a point P moves along the circumference of a circle
at a constant speed, then the central angle 𝜃 that is
formed with the terminal side passing through the
point P also changes over some time t at a constant
speed. The angular speed 𝜔 (omega) is given by
𝜔 =
𝜃
𝑡
where 𝜃 is in radians
20. RELATIONSHIP BETWEEN LINEAR and
ANGULAR SPEEDS
If a point P moves at a constant speed along the circumference
of a circle with radius r , then the linear speed v and
the angular speed 𝜔 are related by
𝒗 = 𝒓𝝎 or 𝜔 =
𝑣
𝑟
Note: The relationship is true only when 𝜃 is in radians.
21. LINEAR and ANGULAR SPEED
Examples:
1. The planet Jupiter rotates every 9.9 hours and has a diameter
of 88,846 miles. If you’re standing on its equator, how fast
are you travelling?
2. Some people still have their phonographic collectionsand
play the records on turntables. A phonograph record is a
vinyl disc that rotates on the turntable. If a 12-inch diameter
record rotates at 33
1
3
revolutions per minute, what is the
angular speed in radians per minute?
22. LINEAR and ANGULAR SPEED
3. How fast is a bicyclist traveling in miles per hour if his tires
are 27 inches in diameter and his angular speed is 5𝜋
radians per second?
4. If a 2-inch diameter pulley that is being driven by an electric
motor and running at 1600 revolutions per minute is
connected by a belt to a 5-inch diameter pulley to drive a
saw, what is the speed of the saw in revolutions per minute?
23. LINEAR and ANGULAR SPEED
5. Two pulleys, one 6 in. and the other 2 ft. in diameter, are
connected by a belt. The larger pulley revolves at the rate of
60 rpm. Find the linear velocity in ft/min and calculate the
angular velocity of the smaller pulley in rad/min.
6. The earth rotates about its axis once every 23 hrs 56 mins 4
secs, and the radius of the earth is 3960 mi. Find the linear
speed of a point on the equator in mi/hr.