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
This article is all about what AI trends will emerge in the field of creative operations in 2024. All the marketers and brand builders should be aware of these trends for their further use and save themselves some time!
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.
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
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
This article is all about what AI trends will emerge in the field of creative operations in 2024. All the marketers and brand builders should be aware of these trends for their further use and save themselves some time!
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.
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
Beginners Guide to TikTok for Search - Rachel Pearson - We are Tilt __ Bright...
Luonggiac chuong3
1. LƯỢNG GIÁC
CHÖÔNG III: PHÖÔNG TRÌNH BAÄC HAI VÔÙI CAÙC HAØM SOÁ LÖÔÏNG GIAÙC
( )
( )
( )
( )
+ + = ≠
+ + = ≠
+ = = ≠
+ + =
2
2
2
2
a sin u b sin u c 0 a 0
a cos u b cos u c 0 a 0
atg u btgu c 0 a 0
a cot g u b cot gu c 0 a 0≠
Caùch giaûi:
Ñaët : hay vôùit sinu= t cosu= t 1≤
(ñieàu kieänt tgu= u k
2
π
≠ + π)
(ñieàu kieänt cot gu= u k≠ π )
Caùc phöông trình treân thaønh: 2
at bt c 0+ + =
Giaûi phöông trình tìm ñöôïc t, so vôùi ñieàu kieän ñeå nhaän nghieäm t.
Töø ñoù giaûi phöông trình löôïng giaùc cô baûn tìm ñöôïc u.
Baøi 56: (Ñeà thi tuyeån sinh Ñaïi hoïc khoái A, naêm 2002)
Tìm caùc nghieäm treân ( cuûa phöông trình)0,2π
( )
cos3x sin 3x
5 sin x 3 cos2x *
1 2sin 2x
+⎛ ⎞
+ = +⎜ ⎟
+⎝ ⎠
Ñieàu kieän:
1
sin 2x
2
≠ −
Ta coù: ( ) ( )3 3
sin 3x cos3x 3sin x 4sin x 4 cos x 3cos x+ = − + −
( ) ( )
( ) ( )
( )( )
3 3
2 2
3 cos x sin x 4 cos x sin x
cos x sin x 3 4 cos x cos xsin x sin x
cos x sin x 1 2sin 2x
= − − + −
⎡ ⎤= − − + + +
⎣ ⎦
= − +
Luùc ñoù: (*) ( ) ( )2
5 sin x cos x sin x 3 2cos x 1⎡ ⎤⇔ + − = +⎣ ⎦ −
1
do sin 2x
2
⎛ ⎞
≠ −⎜ ⎟
⎝ ⎠
2
2cos x 5cos x 2 0⇔ − + =
2. ( )
1
cos x
2
cos x 2 loaïi
⎡
=⎢⇔
⎢
=⎢⎣
x
3
π
⇔ = ± + πk2 (nhaän do
3 1
sin 2x
2 2
= ± ≠ − )
Do ( )x 0,2∈ π neân
5
x x
3 3
π π
= ∨ =
Baøi 57: (Ñeà thi tuyeån sinh Ñaïi hoïc khoái A, naêm 2005)
Giaûi phöông trình: ( )2 2
cos 3x.cos2x cos x 0 *− =
Ta coù: (*)
1 cos6x 1 cos2x
.cos2x 0
2 2
+ +
⇔ − =
cos6x.cos2x 1 0⇔ − = (**)
Caùch 1: (**) ( )3
4 cos 2x 3cos2x cos2x 1 0⇔ − − =
=4 2
4 cos 2x 3cos 2x 1 0⇔ − −
( )
2
2
cos 2x 1
1
cos 2x voâ nghieäm
4
⎡ =
⎢⇔
⎢ = −
⎢⎣
( )
sin 2x 0
k
2x k x k Z
2
⇔ =
π
⇔ = π ⇔ = ∈
Caùch 2: (**) ( )
1
cos8x cos4x 1 0
2
⇔ + − =
( )
2
cos8x cos4x 2 0
2cos 4x cos4x 3 0
cos4x 1
3
cos4x loaïi
2
⇔ + − =
⇔ + −
=⎡
⎢⇔
⎢ = −
⎣
=
( )
k
4x k2 x k Z
2
π
⇔ = π ⇔ = ∈
Caùch 3: phöông trình löôïng giaùc khoâng maãu möïc:
(**) ⇔
cos6x cos2x 1
cos6x cos2x 1
= =⎡
⎢ = = −⎣
Caùch 4: + − = ⇔ +cos 8x cos 4x 2 0 cos 8x cos 4x 2=
⇔ = =cos 8x cos 4x 1 ⇔ =cos 4x 1
Baøi 58: (Ñeà thi tuyeån sinh Ñaïi hoïc khoái D, naêm 2005)
Giaûi phöông trình: 4 4 3
cos x sin x cos x sin 3x 0
4 4
π π⎛ ⎞ ⎛ ⎞
+ + − − −⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠ 2
=
3. Ta coù:
(*)
( )
22 2 2 2 1 3
sin x cos x 2sin x cos x sin 4x sin 2x 0
2 2
⎡ ⎤π⎛ ⎞
⇔ + − + − + −⎜ ⎟⎢ ⎥
⎝ ⎠⎣ ⎦ 2
=
[ ]21 1 3
1 sin 2x cos4x sin 2x 0
2 2 2
⇔ − + − + − =
( )2 21 1 1 1
sin 2x 1 2sin 2x sin 2x 0
2 2 2 2
⇔ − − − + − =
2
sin 2x sin 2x 2 0⇔ + − =
( )
sin 2x 1
sin 2x 2 loaïi
=⎡
⇔ ⎢
= −⎣
π
⇔ = + π ∈
π
⇔ = + π ∈
2x k2 , k
2
x k , k
4
Baøi 59: (Ñeà th ïc khoái B, naêm 2004)i tuyeån sinh Ñaïi ho
( ) (− = − 2
5sin x 2 3 1 sinx tg x *)Giaûi phöông trình:
Khi ñoù: (*)
cos x 0 sin x 1≠ ⇔ ≠ ±Ñieàu kieän:
( )
2
2
sin x
5sin x 2 3 1 sin x
cos x
⇔ − = −
( )
2
2
sin x
5sin x 2 3 1 sin x
1 sin x
⇔ − = −
−
2
3sin x
5sin x 2
1 sin x
⇔ − =
+
2
2sin x 3sin x 2 0⇔ + − =
( )
( )
1
sin x nhaändosin x 1
2
sin x 2 voâ nghieäm
⎡
= ≠⎢⇔
⎢
= −⎢⎣
±
( )
5
x k2 x k2 k
6 6
π π
⇔ = + π ∨ = + π ∈ Z
( )
1 1
2sin 3x 2cos3x *
sin x cos x
− = +Baøi 60: Giaûi phöông trình:
Luùc ñoù: (*)
Ñieàu kieän: sin2x 0≠
( )
1 1
2 sin 3x cos3x
sin x cos x
⇔ − = +
4. ( ) ( )3 3 1 1
2 3 sin x cos x 4 sin x cos x
sin x cos x
⎡ ⎤⇔ + − + = +
⎣ ⎦
( ) ( )2 2 sin x cos x
2 sin x cos x 3 4 sin x sin x cos x cos x
sin x cos x
+⎡ ⎤⇔ + − − + =
⎣ ⎦
( )
1
sin x cos x 2 8sin x cos x 0
sin x cos x
⎡ ⎤
⇔ + − + − =⎢ ⎥
⎣ ⎦
( )
2
sin x cos x 4sin 2x 2 0
sin 2x
⎡ ⎤
⇔ + − −⎢ ⎥
⎣ ⎦
=
( )2
tgx 1sin x cos x 0
nhaän so vôùiñieàu kieän1
sin 2x 1 sin 2x4sin 2x 2sin 2x 2 0
2
= −⎡+ =⎡ ⎢⇔ ⇔ −⎢ ⎢ = ∨ =− − =⎣ ⎣
π π π π
⇔ = − + π ∨ = + π ∨ = − + π ∨ = + π ∈
7
x k 2x k2 2x k2 2x k2 , k
4 2 6 6
π π π
⇔ = ± + π ∨ = − + π ∨ = + π ∈
7
x k x k x k , k
4 12 12
( ) ( )
+ − −
=
+
2
cos x 2sin x 3 2 2 cos x 1
1 *
1 sin 2x
Baøi 61: Giaûi phöông trình:
sin 2x 1 x m
4
π
≠ − ⇔ ≠ − + πÑieàu kieän:
Luùc ñoù:
(*) 2
2sin x cos x 3 2 cos x 2cos x 1 1 sin 2x⇔ + − − = +
2
2cos x 3 2 cos x 2 0⇔ − + =
( )⇔ = =
2
cos x hay cos x 2 voâ nghieäm
2
( )
x k2
4
x k '2 loaïi do ñieàu kieän
4
π⎡
= + π⎢
⇔ ⎢
π⎢ = − + π
⎢⎣
x k2
4
⇔ = + π
π
Baøi 62: Giaûi phöông trình:
( )
x 3x x 3x 1
cos x.cos .cos sin xsin sin *
2 2 2 2 2
− =
Ta coù: (*) ( ) ( )
1 1
cos x cos2x cos x sin x cos2x cos x
2 2
1
2
⇔ + + − =
2
cos x.cos2x cos x sin x cos2x sin x cos x 1⇔ + + − =
cos x⇔ + = − +( ) 2
cos2x cos x sin x 1 cos x sin x
( ) ( )cos2x cos x sin x sin x sin x cos x⇔ + = +
5. ( )( ) ( )cos x sin x cos2x sin x 0 * *⇔ + − =
( )( )2
cos x sin x 1 2sin x sin x 0⇔ + − − =
2
cos x sin x
2sin x sin x 1 0
= −⎡
⇔ ⎢
+ − =⎣
tgx 1
sin x 1
1
sin x
2
⎡
⎢ = −
⎢
⇔ =⎢
⎢
=⎢
⎣
− ( )
x k
4
x k2 k
2
5
x k2 x k2
6 6
π⎡
= − + π⎢
⎢
π⎢⇔ = − + π ∈
⎢
⎢ π π⎢ = + π ∨ = + π
⎢⎣
Z
Caùch khaùc: (**) tgx 1 cos2x sin x cos x
2
π⎛ ⎞
⇔ = − ∨ = = −⎜ ⎟
⎝ ⎠
( )3
4 cos x 3 2 sin 2x 8cos x *+ =Baøi 63: Giaûi phöông trình:
Ta coù: (*) 3
4 cos x 6 2 sin x cos x 8cos x 0⇔ + − =
( )2
cos x 2cos x 3 2 sin x 4 0⇔ + − =
( )2
cos x 2 1 sin x 3 2 sin x 4 0⎡ ⎤⇔ − + −
⎣ ⎦
=
2
cos x 0 2sin x 3 2 sin x 2 0⇔ = ∨ − + =
( )
cos x 0
2
sin x
2
sin x 2 voâ nghieäm
=⎡
⎢
⎢⇔ =
⎢
⎢
=⎢⎣
2
x k sin x sin
2 2
π π
⇔ = + π ∨ = =
4
( )
3
x k x k2 x k2 k
2 4 4
π π π
⇔ = + π ∨ = + π ∨ = + π ∈ Z
Baøi 64: Giaûi phöông trình:
( )cos 2x cos 2x 4sin x 2 2 1 sin x *
4 4
π π⎛ ⎞ ⎛ ⎞
+ + − + = + −⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
( )
(*) ( )2cos2x.cos 4sin x 2 2 1 sin x
4
π
⇔ + = + −
( ) ( )
( )
2
2
2 1 2sin x 4 2 sin x 2 2 0
2 2 sin x 4 2 sin x 2 0
⇔ − + + − − =
⇔ − + + =
6. ( )⇔ − + + =2
2sin x 2 2 1 sin x 2 0
( )⎡
⎢
si =
⇔ ⎢ =⎢⎣
n x 2 loaïi
1
sin x
2
π π
⇔ = + π = + π ∈
5
x k2 hay x k2 , k
6 6
Baøi 65 ( ) ( )+2
g x 2 2 = +2
3cot sin x 2 3 2 cos x *: Giaûi phöông trình :
Ñieàu kieän:
(*)
sin x 0 cos x 1≠ ⇔ ≠ ±
Chia hai veá (*) cho 2
sin x ta ñöôïc:
( )
2
4 2
cos x cos x
3 2 2 2 3 2
sin x sin x
⇔ + = + vaø sin x 0≠
2
cos x
t
sin x
=Ñaët ta ñöôïc phöông trình:
( )2
3t 2 t 2− + +2 3 2 0
2
t 2 t
3
=
⇔ = ∨ =
* Vôùi
2
t
3
= ta coù: 2
cos x 2
3sin x
=
( )
( )
(co nhaän 1
⎢⎣
)
2
2
3cos x 2 1 cos x
2cos x 3cos x 2 0
cos x 2 loaïi
1
s x do cos x
2
⇔ = −
⇔ + − =
⎡ = −
⎢⇔
⎢ = ≠ ±
( )x k2 k
3
π
⇔ = ± + π ∈ Z
* Vôùi t 2= ta coù: =2
cos x
2
sin x
( )
( )
( )
⇔ = −
⇔ + − =
⎡ = −
⎢
⇔ ⎢
= ≠ ±⎢
⎣
π
⇔ = ± + π ∈x k2 , k
2
2
cos x 2 1 cos x
2 cos x cos x 2 0
cos x 2 loaïi
2
cos x nhaän do cos x 1
2
4
Baøi 66: Giaûi phöông trình: ( )
+ − −
=
2 2
4 sin 2x 6sin x 9 3cos 2x
0 *
cos x
7. Ñieàu kieän:
Luùc ñoù:
(*) =
≠cos x 0
2 2
4sin 2x 6sin x 9 3cos2x 0⇔ + − −
( ) ( )2
2
4 1 cos 2x 3 1 cos2x 9 3cos2x 0
4 cos 2x 6cos2x 2 0
1
cos2x 1 cos2x
2
⇔ − + − − − =
⇔ + + =
⇔ = − ∨ = −
2 2 1
2cos x 1 1 2cos x 1
2
⇔ − = − ∨ − = −
( )
( )
( )
cos x 0 loaïi doñieàu kieän
1
cos x nhaän do cos x 0
2
2
x k2 x
3
⇔ = ± + π ∨ k2 k Z
3
⎡ =
⎢⇔ ⎢ = ± ≠
π π
= ± + π ∈
⎢⎣
( )
1 2
f x sin x sin3x sin5x
3 5
= + +Baøi 67: Cho
( )f ' x 0=Giaûi phöông trình:
Ta coù:
=
( )f ' x 0=
( ) ( )
( ) ( )3 2
cos x cos3x 2cos5x 0
cos x cos5x cos3x cos5x 0
2cos3x cos2x 2cos4x cos x 0
4 cos x 3cos x cos2x 2cos 2x 1 cos x 0
⇔ + + =
⇔ + + + =
⇔ + =
⇔ − + −
( )
( )
⎡ ⎤⇔ − + −
⎣ ⎦
⎡⎡ ⎤+ − + − =⎣ ⎦⇔ ⎢
=⎢⎣
⎡ − − =
⇔ ⎢
=⎣
±
⇔ = ∨ =
2 2
2
2
4 cos x 3 cos 2x 2 cos 2x 1 cos x 0
2 1 cos 2x 3 cos 2x 2cos 2x 1 0
cos x 0
4 cos 2x cos 2x 1 0
cos x 0
1 17
cos 2x cos x 0
8
=
( )
1 17 1 17
cos2x cos cos2x cos cos x 0
8 8
x k x k x k k Z
2 2 2
+ −
⇔ = = α ∨ = = β ∨ =
α β π
⇔ = ± + π ∨ = ± + π ∨ = + π ∈
8. ( )8 8 217
sin x cos x cos 2x *
16
+ =Baøi 68: Giaûi phöông trình:
Ta coù:
( )
( )
28 8 4 4 4 4
22
2 2 2 2 4
2
2 4
2 4
sin x cos x sin x cos x 2sin x cos x
1
sin x cos x 2sin x cos x sin 2x
8
1 1
1 sin 2x sin 2x
2 8
1
1 sin 2x sin 2x
8
+ = + −
⎡ ⎤= + − −⎢ ⎥⎣ ⎦
⎛ ⎞
= − −⎜ ⎟
⎝ ⎠
= − +
Do ñoù:
( ) ( )
( )
( )
( ) ( )
⎛ ⎞
⇔ − + = −⎜ ⎟
⎝ ⎠
⇔ + − =
⎡ = −
⎢⇔ ⇔ −⎢ =⎢
=
π
⇔ = ⇔ = + ∈
2 4 2
4 2
2
2
1
* 16 1 sin 2x sin 2x 17 1 sin 2x
8
2sin 2x sin 2x 1 0
sin 2x 1 loaïi
1 1
1 cos 4x1 2 2sin 2x
cos 4x 0 x 2k 1 , k Z
8
Baøi 69
⎣ 2
( )35x x
sin 5cos x.sin *
2 2
=: Giaûi phöông trình:
Nhaän xeùt thaáy:
x
cos 0 x k2 cos x 1
2
= ⇔ = π + π ⇔ = −
Thay vaøo (*) ta ñöôïc:
π⎛ ⎞ ⎛
+ π = − + π⎜ ⎟ ⎜
⎝ ⎠ ⎝
5
sin 5k 5.sin k
2 2
π ⎞
⎟
⎠
, khoâng thoûa k∀
x
cos
2
Do khoâng laø nghieäm cuûa (*) neân:
( ) ⇔ = 25x x x x
* sin .cos 5cos x.sin cos
2 2 2 2
vaø
x
cos 0
2
≠
( ) 31 5
sin 3x sin 2x cos x.sin x
2 2
⇔ + = vaø ≠
x
cos 0
vaø
2
3 3
3sin x 4sin x 2sin x cos x 5cos x.sin x⇔ − + = ≠
x
cos 0
2
2 3
x
cos 0
2
3 4sin x 2cos x 5cos x sin x 0
⎧
≠⎪
⇔ ⎨
⎪ − + = ∨⎩ =
9. 3 2
x
cos 0
2
x
5cos x 4 cos x 2cos x 1 0 sin 0
2
⎧
≠⎪⎪
⇔ ⎨
⎪ − − + = ∨
⎪⎩
=
( )( )2
cos x 1
x
cos x 1 5cos x cos x 1 0 sin 0
2
≠ −⎧
⎪
⇔ ⎨
− + − = ∨ =⎪⎩
≠ −⎧
⎪
⎡⎪
⎢ =⎪
⎢⎪
⇔ − +⎨⎢ = = α⎪⎢
⎪⎢
− −⎪⎢
= = β
⎣⎩
cos x 1
cos x 1
1 21
cos x cos
10
1
cos
10⎪⎢
1 2
cos x
( )⇔ = π = ±α + π = ±β + π ∈x k2 hay x k2 hay x k2 , k Z
( ) ( )2
sin 2x cot gx tg2x 4 cos x *+ =Baøi 70: Giaûi phöông trình:
ieàu kieän: vaø cos2x 1Ñ 0cos2x 0≠ sin x 0 cos2x≠ ⇔ ≠ ∧ ≠
Ta coù:
cos x sin2x
cot gx tg2x
sin x cos2x
+ = +
cos2x cos x sin 2xsin x
sin x cos2x
cos x
sin x cos2x
+
=
=
2cos x
2sin x.cos x 4 cos x
sin x cos2x
⎛ ⎞
⇔ =⎜ ⎟
⎝ ⎠
Luùc ñoù: (*)
( ) ( )
( )
( )
⇔ =
⇔ + = +
⇔ + = =
⇔ = − ∨ = ≠ ≠
2
2cos x
2cos x
cos 2x
cos 2x 1 2cos 2x cos 2x 1
cos 2x 1 0 hay 1 2cos 2x
1
cos 2x 1 cos 2x nhaän do cos 2x 0 vaø cos 2x 1
2
π
⇔ = π + π ∨ = ± + π ∈
π π
⇔ = + π ∨ = ± + π ∈
2x k2 2x k2 , k
3
x k x k , k
Baøi 71
2 6
( )2 6x 8x
2cos 1 3cos *
5 5
+ =: Giaûi phöông trình:
10. ⎛ ⎞ ⎛ ⎞
⇔ + + =⎜ ⎟ ⎜
⎝ ⎠ ⎝
212x 4x
1 cos 1 3 2cos 1
5 5
Ta coù : (*) − ⎟
⎠
⎛ ⎞
⇔ + − = ⎜ ⎟
⎝ ⎠
3 24x 4x 4x
2 4cos 3cos 3 2cos 1
5 5 5
−
Ñaët ( )4
t cos x ñieàu kieän t 1
5
= ≤
Ta coù phöông trình :
( )( )
( )
3 2
3 2
2
4t 3t 2 6t 3
4t⇔ 6t 3t 5 0
t 1 4t 2t 5 0
1 21 1 21
t 1 t t loïai
4 4
− + = −
− − + =
⇔ − − − =
− +
⇔ = ∨ = ∨ =
Vaäy
( )
• = ⇔ = π
π
⇔ = ∈
4x 4x
cos 1 2k
5 5
5k
x k
2
Z
( )
( )
4x 1 21
cos cos vôùi 0 2
5 4
4x
2
5
5 5
x , Z
4 2
−
• = = α < α < π
⇔ = ±α + π
α π
⇔ = ± + ∈
l
l
l
Baøi 72 ( )3
tg x tgx 1 *
4
π⎛ ⎞
− = −⎜ ⎟
⎝ ⎠
: Giaûi phöông trình
t x x t
4 4
π π
= − ⇔ = +Ñaët
3 1 tgt
tg t tg t 1 1 vôùi cost 0 tgt 1
4 1 tgt
π +⎛ ⎞
= + − = − ≠ ∧⎜ ⎟ −⎝ ⎠
(*) thaønh : ≠
⇔ =
−
3 2tgt
tg t
1 tgt
( )
)( )
( )
(
3 4
3 2
2
tg t tg t 2tgt
tgt tg t tg t 2 0
t 1 tg t 2tgt 2 0
tgt 0 tgt 1 nhaän so ñi àu kieän
t k t k ,k
4
⇔ − =
⇔ − + =
+ − + =
⇔ = ∨ = −
π
⇔ = π ∨ = − + π ∈¢
Vaäy (*)
tgt tg⇔
e
11. x k hay x
4
⇔ = + π = k ,k
π
π ∈¢
Baøi 73
4 4
4sin 2x cos 2x
cos 4x (*)
tg x tg x
4 4
+
=
π π⎛ ⎞ ⎛ ⎞
− +⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
: Giaûi phöông trình
Ñieàu kieän
sin x cos x 0 sin 2x 0
4 4 2
sin
π π⎛⎪
⎜ x cos x 0 sin 2x 0
4 4 2
⎧ ⎧π π π⎛ ⎞ ⎛ ⎞ ⎛ ⎞
− − ≠ − ≠⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎪ ⎪
⎪ ⎝ ⎠ ⎝ ⎠ ⎪ ⎝ ⎠
⎨ ⎨
π⎞ ⎛ ⎞ ⎛ ⎞⎪+ + ≠ + ≠⎟ ⎜ ⎟ ⎜ ⎟⎪ ⎪⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎩ ⎩
±
Do :
⇔
cos2x 0 sin2x 1⇔ ≠ ⇔ ≠
1 tgx 1 tgx
tg x tg x . 1
4 4 1 tgx 1 tgx
π π − +⎛ ⎞ ⎛ ⎞
− + =⎜ ⎟ ⎜ ⎟
+ −⎝ ⎠ ⎝ ⎠
=
Khi cos2x 0 thì :≠
( )
( )
( )
( )
4 4 4
2 2 4
2 4
2 4
4 2
2
2
2
* sin 2x cos 2x cos 4x
1 2sin 2x cos 2x cos 4x
1
1 sin 4x cos 4x
2
1
1 1 cos 4x cos 4x
2
2cos 4x cos 4x 1 0
cos 4x 1
1 sin 4x 11
cos 4x voâ nghieäm
2
sin 4x 0
2sin2x cos2x 0
sin2x 0 do cos2x 0
2x k ,k x k
⇔ + =
⇔ − =
⇔ − =
⇔ − − =
⇔ − − =
⎡ =
⎢⇔ ⇔
⎢ = −
⎢⎣
⇔ =
⇔ =
⇔ = ≠
⇔ = π ∈ ⇔ =¢
− =
, k
2
π
∈¢
( )4 2
1 2
48 1 cot g2x cot gx 0 *
cos x sin x
− − + = ( )Baøi 74 :Giaûi phöông trình:
Ñieàu kieän :
Ta coù :
sin2x 0≠
12. ( )2 2
cos2x cosx
1 cot g2x cot gx 1 .
sin2x sin x
sin2xsin x cos2x cosx
sin xsin2x
cosx 1
do cosx 0
2sin x cosx 2sin x
+ = +
+
=
= = ≠
Luùc ñoù (*) 4 4
1 1
48 0
cos x sin x
⇔ − − =
4
4 4 4 4
1 1 sin x cos
48
cos x sin x sin x cos x
+
⇔ = + =
4
x
4 4 4 4
4 2 2
4 2
48sin x cos x sin x cos x
3sin 2x 1 2sin x cos x
1
3sin 2x sin 2x 1 0
2
⇔ = +
⇔ = −
⇔ + − =
( )
( )
2
2
2
sin x loïai
3
1
sin x nhaän do 0
2
⎡
= −⎢
⇔ ⎢
⎢ = ≠
⎢⎣
( )
( )
2 2
cos4x 0
2
k
x k Z
8 4
⇔ =
π
π π
⇔ = + ∈
Baøi 75
1 1
1 cos4x⇔ − =
4x k
π
⇔ = +
: Giaûi phöông trình
( ) ( )8 8 10 10 5
sin x cos x 2 sin x cos x cos2x *
4
+ = + +
Ta coù : (*)
( ) ( )8 10 8 10 5
sin x 2sin x cos x 2cos x cos2x
4
⇔ − + − =
( ) ( )
( )
8 2 8 2
8 8
8 8
5
sin x 1 2sin x cos x 1 2cos x cos2x
4
5
sin x.cos2x cos x cos2x cos2x
4
4cos2x sin x cos x 5cos2x
⇔ − − − + =
⇔ − =
⇔ − =
13. ( )
( )( )
8 8
2
2
s2x 0 hay 4 sin x x 5
cos2x 0 hay 4 1 sin 2x 5
2
cos2x 0 hay 2sin 2x 1(Voâ nghieäm )
= =
= − =⎜ ⎟
⎝ ⎠
⇔ = − =
co cos⇔ −
4 4 4 4
cos2x 0 hay 4 sin x cos x sin x cos x 5
1
⇔ = − + =
⎛ ⎞
⇔
2x k ,k
π
⇔ = + π ∈¢
2
k
x ,k
4 2
π π
⇔ = + ∈¢
( )8 8
4 sin x cos x 5− =Caùch khaùc: Ta coù voâ nghieäm
Vì ( )8 8
sin x cos x 1, x− ≤ ∀ neân ( )8 8
4 sin x cos x 4 5, x− ≤ < ∀
Ghi chuù : Khi gaëp phöông trình löôïng giaùc daïng R(tgx, cotgx, sin2x, cos2x, tg2x)
vôùi R haøm höõu tyû thì ñaët t = tgx
Luùc ñoù
2
2 2
2t 2t 1 t
tg2x ,sin2x ,cos2x
1 t 1 t 1 t2
−
= = =
− + +
Baøi 76 : (Ñeå thi tuyeån sinh Ñaïi hoïc khoái A, naêm 2003)
Giaûi phöông trình
( )− = + −
+
2cos 2x 1
cot gx 1 sin x sin 2x *
1 tgx 2
Ñieàu kieän : −
Ñaët t = tgx thì (*) thaønh :
sin2x 0vaø tgx 1≠ ≠
2
22
2 2
1 t
1 1 1 t 1 2t1 t1 1 .
t 1 t 2 1 t 2 1 t
−
⎡ ⎤−+− = + − −⎢ ⎥+ + +⎣ ⎦
( )
( )
( )( ) ( )
( )
( )
( )
2
2 2 2
2
2 2
22
2 2
2t t
do t 1
t 2 t 1 t
t
t 1 t 1 t
1 t 1 t 1 t t
t 1 nhaän do t 11 t 0
1 t 1 t t 2t t 1 0 voâ nghieäm
− ≠ −
+
+
⇔ − + = −
= ≠ −⎡− =⎡
⇔ ⇔ ⎢⎢
+ = − − + =⎢⎣ ⎣
Vaäy (*)
1 t 1 t 1
.
− −
⇔ = +
t 1 1+ +
2
11 t t 2t 1 −− − +
⇔ = =
+
⇔ ( )tgx 1 x k nhaän do sin2x 1 0
4
π
= ⇔ = + π = ≠
Baøi 77 ( )+ =sin 2x 2tgx 3 *: Giaûi phöông trình:
Ñieàu kieän :
aët t = tgx thì (*) thaønh :
cos x 0≠
Ñ
14. 2
2t
2t 3+ =
1 t+
)( )
( )
( )( )
(
( )
⇔ + − + =
+ − =
− − + =
− + =⎣
π
⇔ = ⇔ = + π ∈
2
2
2t 2t 3 1 t 0
4t 3
1 2t t 3 0
2t t 3 0 voâ nghieäm
⇔ −3 2
2t 3t 0
⇔ t
=⎡
⇔ ⎢ 2
t 1
Vaäy (*) tgx 1 x k k Z
4
Baøi 78 : Giaûi phöông trình
( )
2
cot gx tgx 4sin2x *
sin 2x
− + =
sin2x 0≠Ñieàu kieän :
Ñaët 2
2t
t tgx thì : sin 2x do sin 2x 0 neân t 0
1 t
= = ≠
+
≠
(*) thaønh :
2
2
1 8t 1 t 1
t t
t t1 t t
+
− + = = +
+
( )
( )
⇔ =
+
⇔ = ≠
+
⇔ = ⇔ = ± ≠
π⎛ ⎞
⇔ = ±⎜ ⎟
⎝ ⎠
π
⇔ = ± + π ∈
2
2
4
1 do t 0
1 t
t 3 t 3 nhaän do t 0
Vaäy (*) tgx tg
3
x k , k
3
Baøi 79
2
8t
2t
1 t
: Giaûi phöông trình
( )( ) ( )1 tgx 1 sin 2x 1 tgx *− + = +
Ñieàu kieän :
Ñaët = tgx thì (*) thaønh :
cos x 0≠
( ) 2
1 t
⎜ ⎟
+⎝ ⎠
2t⎛
1 t 1 1 t
⎞
− + = +
( )
( )
2
2
1 t 1 t
1 t
− = +
+
( )( ) 2 2
2
t 1
t 1
t 1
1 t
1 t 1 t1
1 t
t 1 t 0
+
⇔
= −⎡
= −⎡
+ ⎢⎢ 1 t
⎢⇔ ⇔−
− = += ⎣⎢ +⎣
⇔ = − ∨ =
15. = −⎡ π
⇔Do ñoù (*) ⇔ = − + π = π ∈⎢ =⎣
tgx 1
x k hay x k , k
tgx 0 4
Baøi 80 ( ): Cho phöông trình ( )1 0 *+ =cos2x 2m 1 cos x m− + +
3
m
2
=a/ Giaûi phöông trình khi
3
,
2 2
π π⎛ ⎞
⎜ ⎟
⎝ ⎠
b/ Tìm m ñeå (*) coù nghieäm treân
( )2
2cos x 2m 1 cos x m 0− + + =Ta coù (*)
[ ]( )
( )
⎧ = ≤⎪
⇔ ⎨
− + + =⎪⎩
2
t cos x t 1
2t 2m 1 t m 0
[ ]( )⎧ = ≤
⎪
⇔ ⎨
= ∨ =⎪
⎩
t cos x t 1
1
t t m
2
a/ =Khi m , phö
2
3
ông trình thaønh
b/
( )
( )
)
[ ]
( ) )
π π⎛ ⎞
∈ = ∈ −⎜ ⎟
⎝ ⎠
= ∉ −
⎞
⇔ ∈ −⎡⎣
⎝ ⎠
1 3
loaïi
3
Khi x , thì cos x t [ 1, 0
2 2
1
Do t 1, 0 neân
2
m 1, 0
2 2
Baøi 81
= ∨ =cos x cos x
2 2
π
⇔ = ± + π ∈x k2 k Z
3
π π⎛
⎜ ⎟
3
* coù nghieäm treân ,
: Cho phöông trình
( )( ) ( )x *2
cos x 1 cos2x m cos x m sin+ − =
a/ Giaûi (*) khi m= -2
2
0,
π⎡
b/ Tìm m sao cho (*) coù ñuùng hai nghieäm treân
3
⎤
⎢ ⎥⎣ ⎦
( ) (( ) )
( ) ( )
)(
2 2
2
2
Ta coù (*) cos x 1 2cos x 1 m cos x m 1 cos x
cos x 1 2cos x 1 m cos x m 1 cos x 0
1 2cos x 1
⇔ + − − = −
⎡ ⎤⇔
( )cos x m 0
+ − − − − =⎣ ⎦
+ −
i m = -2 thì (*) thaønh :
⇔ − =
a/ Kh
16. ( )( )
( )
+ + =
⇔
⇔ = π + π ∈
π⎡ ⎤ ⎡
∈ = ∈⎢ ⎥ ⎢⎣ ⎦ ⎣
2
cos x 1 2 cos x 1 0
cosx = -1
x k2 k Z
2 1
b / Khi x 0, thì cos x t ,1
3 2
⎤
− ⎥⎦
Nhaän xeùt raèng vôùi moãi t treân
1
,1
2
⎡ ⎤
−⎢ ⎥⎣ ⎦
ta chæ tìm ñöôïc duy nhaát moät x treân
2
0,
π⎡ ⎤
⎢ ⎥3⎣ ⎦
ùng hai ghieäm treân
1
,1
2
⎡ ⎤
−⎢ ⎥⎣ ⎦
Yeâu caàu baøi toaùn 2
2t 1 m 0⇔ − − = coù ñu n
Xeùt ( ) ( )2
y 2t 1 P vaø y m d= − =
Ta coù y’ = 4t
2
0,
3
π⎡ ⎤
⎢ ⎥⎣ ⎦
Vaäy (*) coù ñuùng hai nghieäm treân
1
,1
2
⎡ ⎤
−⎢ ⎥⎣ ⎦
⇔ (d) caét (P) taïi hai ñieåm phaân bieät treân
1
1 m
2
− < ≤⇔
Baøi 82 : Cho phöông trình ( ) ( )2 2
1 a tg− x 1 3a 0 1
cos x
− + + =
1
a
2
=a/ Giaûi (1) khi
0,
2
π⎛ ⎞
⎜ ⎟
⎝ ⎠
b/ Tìm a ñeå (1) coù nhieàu hôn moät nghieäm treân
Ñieàu kieän : cos x 0 x k
2
π
≠ ⇔ ≠ + π
17. ( ) ( ) ( )
( )( ) ( )
( ) ( )
( ) ( )
2 2
2 2
2
4a cos x 2cos x 1 a 0⇔ − + − =
2
1 1 a sin x 2cos x 1 3a cos x 0
1 a 1 cos x 2cos x 1 3a cos x 0
a 4 cos x 1 2cos x 1 0
2cos x 1 a 2cos x 1 1 0
⇔ − − + + =
⇔ − − − + + =
⇔ − − − =
⇔ − + − =⎡ ⎤⎣ ⎦
a/ Khi
1
a
2
= thì (1) thaønh : ( )
1
2cos x 1 cos x 0
2
⎛ ⎞
− − =⎜ ⎟
⎝ ⎠
( )
( )
1
cos x cos nhaändo cos x 0
2 3
x k2 k Z
3
π
⇔ = = ≠
π
⇔ = ± + π ∈
b/ Khi x 0,
π⎛ ⎞
∈ thì
2
⎜ ⎟
⎝ ⎠
( )t 0,1= ∈cos x
( )
( )
1
cos x t 0,1
2
2a cos x 1 a 2
⎡
= = ∈⎢⇔
⎢
= −
Ta coù : (1)
⎢⎣
Yeâu caàu baøi toaùn ⇔ (2) coù nghieäm treân ( )
a 0
1 1 a
0,1
⎧ ⎫
0 1
2 2a
1 a 1
2a 2
⎧
⎪ ≠
⎪
−⎪
⇔ < <⎨
⎩ ⎭ ⎪
−⎪
≠⎪⎩
⎨ ⎬
a 0≠⎧
( )
01 a
0
⎪ <−⎪ >
a 1 1
a 1
12a 3
a 0 a
1 3a 13
0 a
12a 2
a
2 1 a 2a 2
⎧
⎪ < ⎧
< <⎪⎪ ⎪ ⎪
⇔ ⇔ < ∨ > ⇔⎨ ⎨ ⎨
−⎪ ⎪ ⎪< ≠
⎪⎪ ⎪ ⎩
≠⎪ ⎪− ≠ ⎩⎩
Caùch khaùc
⎪
cos x
1
, ñieàu kieän ; pt thaønhu ≥1: daët u =
( ) ( )−1 a − − + + = ⇔ − − + =2 2
( u 1) 2u 1 3a 0 1 a u 2u 4a 0
Baøi 83
⇔ − − − =( u 2)[ (1 a)u 2a ] 0
( )cos4x 6sin x cos x m 1+ =: Cho phöông trình :
a/ Giaûi (1) khi m = 1
0,
4
π⎡ ⎤
⎢ ⎥⎣ ⎦
(1) coù hai nghieä phaân bieät treânb/ Tìm m ñeå m
⇔ − +2
1 2sin 2x 3sin 2x mTa coù : (1) =
18. ( )
( )2
t sin 2x t 1
2t 3t m 1 0 2
⎧ = ≤⎪
⇔ ⎨
− + − =⎪⎩
a/ Khi m = 1 thì (1) thaønh
( ) ( )
( )2
t sin 2x t 1
t sin 2x t 1
3
t 0 t loaïi2t 3t 0
2
= ∨ =− =⎪ ⎪⎩
⎩
k
sin 2x 0 x
2
⎧ = ≤⎧ = ≤⎪ ⎪
⇔⎨ ⎨
π
⇔ = ⇔ =
[b/ Khi ]∈
⎣ ⎦
hì sin 2x t 0,1
4
Nhaän thaáy raèng moãi t tìm ñöôïc treân
π⎡ ⎤
∈ =⎢ ⎥x 0, t
[ ]0,1 ta chæ tìm ñöôïc duy nhaát moät
x 0,
4
π⎡ ⎤
∈ ⎢ ⎥⎣ ⎦
Ta coù : (2) ⇔
Xeùt
2
2t 3t 1 m− + + =
[ ]2
y 2t 3t 1treân 0,1= − + +
Thì y ' 4t 3= − +
[ ]0,1Yeâu caàu baøi toaùn ⇔ (d) y = m caét taïi hai ñieåm phaân bieät treân
17
2 m
8
⇔ ≤ <
Caùch khaùc :ñaët . Vì a = 2 > 0, neân ta coù
Yeâu caàu baøi toaùn ⇔
= − + −2
f(x) 2t 3t m 1
( )f
Δ =⎧
⎪
( )
m
m
f m
S
− >
= − ≥⎪⎪
⎨ = − ≥
⎪
⎪ ≤ = ≤
⎪⎩
17 8 0
1
1 2
3
0 1
2 4
0 0
0
17
2 m
8
⇔ ≤ <
Baøi 84 : Cho phöông trình
( )5 5 2
4 cos x.sin x 4sin x cos x sin 4x m 1− = +
a/ Bieát raèng laø nghieäm cuûa (1). Haõy giaûi (1) trong tröôøng hôïp ñoù.x = π
x
8
π
= −b/ Cho bieát laø moät nghieäm cuûa (1). Haõy tìm taát caû nghieäm cuûa (1) thoûa
4 2
x 3x 2− + < 0
19. ( )
( )( )
( )
4 4 2
2 2 2 2 2
2
2
(1) 4sin x cos x cos x sin x sin 4x m
2sin 2x cos x sin x cos x sin x sin 4x m
2sin 2x.cos2x sin 4x m
sin 4x sin 4x m 0 1
⇔ − = +
⇔ − + = +
⇔ = +
⇔ − + =
a/ laø nghieäm cuûa (1) = 0
Luùc ñoù (1)
x = π 2
sin 4 sin4 m⇒ π − π +
m 0⇒ =
( )sin 4x 1 sin 4x 0⇔ − =
( )
⇔ = ∨ =
π
⇔ = π ∨ = + π
π π π
⇔ = ∨ = + ∈
sin 4x 0 sin 4x 1
4x k 4x k2
2
k k
x x k Z
4 8 2
b/
2 2
4 2
2
t x 0 t x 0
x 3x 2 0
1 t 2t 3t 2 0
⎧ = ≥ ⎧ = ≥⎪
− + < ⇔ ⇔⎨ ⎨
< <− + < ⎩⎪⎩
( )
2
1 x 2 1 x 2
2 x 1 1 x 2 *
<
⇔ − < < − ∨ < <
⇔ < < ⇔ <
π π⎛ ⎞
= − = − = −⎜ ⎟
⎝ ⎠
x thì sin 4x sin 1
8 2
( )x laø nghieämcuûa 1 1 1 m 0
8
m 2
π
= − ⇒ + + =
⇒ = −
2
sin 4x sin4x 2 0− − =Luùc ñoù (1) thaønh :
( )
( )
( )
2
t sin 4x vôùi t 1
t t 2 0
t sin 4x vôùi t 1
t 1 t 2 loaïi
⎧ = ≤⎪
⇔ ⎨
− − =⎪⎩
⎧ = ≤⎪
⇔ ⎨
= − ∨ =⎪⎩
sin 4x 1
4x k2
2
k
x
8 2
Keát hôïp vôùi ñi
⇔ = −
π
⇔ = − + π
π π
⇔ = − +
eàu kieän (*) suy ra k = 1
ghieäm
3
x
8 2 8
π π π
= − + = thoûa 04 2
x 3x 2− + <Vaäy (1) coù n
Baøi 85 : Tìm a ñeå hai phöông trình sau töông ñöông
( )
( )( ) (2
)
1 cos2x cos3x 1
4 cos x cos3x a cos x 4 a 1 cos2x 2
+ +
− = + − +
2cos x.cos2x =
20. ( )
( )
2
Ta coù : (1) cos3x cos x 1 cos2x cos3x
cos x 1 2cos x 1
cos x 1 2cos x 0
1
cos x 0 cos x
2
⇔ + = + +
⇔ = + −
⇔ − =
⇔ = ∨ =
( )⇔ − − =2 3
Ta coù : (2) 4 cos x 4 cos x 3cos x a co ( )
( ) ( )
( )
+ −
+ − − =
⇔ ⎢
+ − + − =⎢⎣
2
2
2
s x 4 a 2 cos x
4 2a cos x a 3 cos x 0
0
4 cos x 2 2 a cos x a 3 0
⇔ 3
4 cos x
=⎡cos x
[ ]⎛ ⎞
⇔ = − + − =⎜ ⎟
⎝ ⎠
1
cos x 0 hay cos x 2 cos x 3 a 0
2
−
⇔ = ∨ = ∨ =
1 a
cos x 0 cos x cos x
2 2
3
Vaäy yeâu caàu baøi toaùn
a 3
0
2 a 3
a 3 1
a 4
2 2
a 1 a 5
a 3 a 3
1 1
−⎡
=⎢
=⎡⎢
− ⎢⎢⇔ = ⇔ =⎢⎢
⎢ <
2 2
∨ >⎢ ⎣− −⎢ < − ∨ >
Baøi 86
⎢⎣
: Cho phöông trình : cos4x = cos2
3x + asin2
x (*)
a/ Giaûi phöông trì nh khi a = 1
0,
12
π⎛ ⎞
⎜ ⎟
⎝ ⎠
b/ Tìm a ñeå (*) coù nghieäm treân
( ) ( ) ( )
1 a
* cos4x 1 cos6x 1 cos2x
2 2
⇔ = + + −Ta coù :
( ) ( )
( )
( ) ( )
2 3
2 3
2 2cos⇔ 2x 1 1 4 cos 2x 3cos2x a 1 cos2x
t cos2x t 1
2 2t 1 1 4t 3t a 1 t
− = + − + −
⎧ = ≤⎪
⇔ ⎨
− = + − + −⎪⎩
( )
( )
( )
( )( ) ( ) ( )
3 2
2
t cos2x t 1
4t 4t 3t 3 a 1 t
1 cos2x t 1
t 1 4t 3 a 1 t * *
⎧ = ≤⎪
⇔ ⎨
− + + − = −⎪⎩
⎧ = ≤⎪
⇔ ⎨
− − + = −⎪⎩
a/ Khi a = 1 thì (*) thaønh :
21. ( )
( )( )
( )
2
t cos2x t 1
t 1 4t 4 0 t 1
t cos2x t 1⎧ = ≤ ⎧ = ≤
− − + = = ±⎪⎪ ⎩⎩
⎪ ⎪
⇔⎨ ⎨
( )
⇔ = ± ⇔ =
π
⇔ = ⇔ = π ⇔ = ∈
2
cos 2x 1 cos 2x 1
k
sin 2x 0 2x k x , k Z
2
3
x 0, 2x 0, .Vaäy c
6
⎛ ⎞
∈ ⎜ ⎟
⎝ ⎠
os2x t ,1
12 2
⎛ ⎞π π⎛ ⎞
∈ ⇔ = ∈ ⎜ ⎟⎜ ⎟ ⎜ ⎟
⎝ ⎠
− + = −
⇔ − = ≠2
b/ Ta coù :
⎝ ⎠
2
(⇔ )( ) ( )
( )
Vaäy (**) t-1 4t 3 a 1 t
4t 3 a do t 1
eùt ( )2 3
y P4t 3 treân ,1
2
⎛ ⎞
= − ⎜ ⎟⎜ ⎟
⎝ ⎠
X
3
y ' 8t 0 t ,1
2
⎛ ⎞
⇒ = > ∀ ∈ ⎜ ⎟⎜ ⎟
⎝ ⎠
ñ ù (*) coù nghieäm treâ ( ) ( )
⎛ ⎞π⎛ ⎞
⇔ = ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
3
0, d : y a caét P treân ,1
2 2
Do o n
( )
3
y a y
2
0 a 1
1<⎜ ⎟⎜ ⎟
⎝ ⎠
< <
BAØI TAÄP
⎛ ⎞
⇔ <
⇔
ûi ùc phöông trình sau :1. Gia ca
a/ sin4x = tgx
b/ 4 4 4 9
sin
π π
x sin x x sin x
4 4 8
⎛ ⎞ ⎛ ⎞
+ + + − =⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
c/ tgx cot gx 4+ =
d/
( ) 2
sin x 3 2 2cos x 2sin x 1
1
1 sin 2x
− − −
=
−
e/ 4
4 cos x 3 2 sin 2x 8cos x+ =
f/
1 1 2
cos sin 2x sin 4x
+ =
x
g/ sin 2x 2 sin x 1
4
π⎛ ⎞
+ −⎜ ⎟
⎝ ⎠
=
h/ ( ) ( )2 2sin x 1 4 sin x 1 cos 2x sin 2x
4 4
π π⎛ ⎞ ⎛ ⎞
− = − − + − +⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
k/ 24x
cos cos x
3
=
l/
x
tg .cos x sin 2x 0
2
+ =
22. m/ 1 3tgx 2sin2x+ =
n/ cot gx tgx 2tg2x= +
p/ + =2 3x 4x
2cos 1 3cos
5 5
q/ =2
3cos4x 2cos 3x 1−
r/ 2 3x
2cos 1 3cos2x+ =
2
x
s/ cos x tg 1
2
+ =
t/
u/
2
3tg2x 4tg3x tg 3x.tg2x− =
2 3
cos x.cos4x cos2x.cos3x cos 4x
2
+ + =
v/ 2 2 2 2 3
cos x cos 2x cos 3x cos 4x
2
+ + + =
w/
x/
sin4x tgx=
6 6 213
cos x sin x cos 2x
8
+ =
y/
3 x 1 3x
sin
π π⎞ ⎛ ⎞
− = +sin
2
⎛
⎜ ⎟ ⎜ ⎟
⎠
( 1 )
a/ Giaûi phöông trình khi a = 1.
10 2 2 10⎝ ⎠ ⎝
. 6 6
sin x cos x a sin 2x+ =2
1
a
4
≥ )b/ Tìm a ñeå (1) coù nghieäm (ÑS :
3. Cho phöông trình
( )
6 6
2 2
cos x sin x
2mtg2x 1
cos x sin x
+
=
−
a/ Giaûi phöông trình khi m =
1
8
1
m
8
≥b/ Tìm m sao cho (1) coù nghieäm (ÑS : )
.4 Tìm m ñeå phöông trình
x kπsin4x mtgx coù nghieäm= ≠
1
ÑS : m 4
2
⎛ ⎞
− < <⎜ ⎟
⎝ ⎠
5. Tìm m ñeå phöông trình :
coù ñuùng 7 nghieäm treân
cos3x cos2x mcos x 1− + − = 0
,2
2
π⎛ ⎞
− π⎜ ⎟
⎝ ⎠
( )ÑS :1 m 3< <
6. Tìm m ñeå phöông trình :
( ) ( )4 4
4 sin x cos 6 6 2
x 4 sin x c 4x mos x sin− + = coù nghieäm+ −
1
ÑS : m 1
8
⎛ ⎞
− ≤ ≤⎜
⎝
⎟
⎠
23. 7. Cho phöông trình :
2 2 2
6sin x sin x m cos 2x− = (1)
a/ Giaûi phöông trình khi m = 3
b/ Tìm m ñeå (1) coù nghieäm ( )ÑS :m 0≥
8. Tìm m ñeå phöông trình :
( )4 22m 1m
sin x cos4x sin 4x sin x 0
4 4
+
+ + − =
coù hai nghieäm phaân bieät treân ,
4 2
π π⎛ ⎞
⎜ ⎟
⎝ ⎠
1
ÑS :2 5 4 m
2
⎛ ⎞
− < <⎜ ⎟
⎝ ⎠
9. Tìm m ñeå phöông trình :
coù nghieäm( )6 6 4 4
sin x cos x m sin x cos x+ = +
1
ÑS : m 1
2
⎛ ⎞
≤ ≤⎜ ⎟
⎝ ⎠
10. Cho phöông trình :
Tìm a ñeå phöông trình coù nghieäm
2 2
cos4x cos 3x a sin x= +
x 0,
2
π⎛ ⎞
∈ ⎜ ⎟
⎝ ⎠
( )ÑS :0 a 1< <
Th.S Phạm Hồng Danh
TT luyện thi đại học CLC Vĩnh Viễn