SlideShare a Scribd company logo
Holt Algebra 2
UNIT 10.1UNIT 10.1
TRIGONOMETRICTRIGONOMETRIC
IDENTITIESIDENTITIES
Warm Up
Simplify.
1.
2.
cos A
1
Use fundamental trigonometric
identities to simplify and rewrite
expressions and to verify other
identities.
Objective
You can use trigonometric identities to simplify
trigonometric expressions. Recall that an
identity is a mathematical statement that is true
for all values of the variables for which the
statement is defined.
A derivation for a Pythagorean identity is
shown below.
x2
+ y2
= r2
cos2
θ + sin2
θ = 1
Pythagorean Theorem
Divide both sides by r2
.
Substitute cos θ for and
sin θ for
To prove that an equation is an identity, alter one
side of the equation until it is the same as the
other side. Justify your steps by using the
fundamental identities.
Example 1A: Proving Trigonometric Identities
Prove each trigonometric identity.
Choose the right-hand side
to modify.
Reciprocal identities.
Simplify.
Ratio identity.
Example 1B: Proving Trigonometric Identities
Prove each trigonometric identity.
1 – cot θ = 1 + cot(–θ)
= 1 + (–cotθ)
= 1 – cotθ
Choose the right-hand side
to modify.
Reciprocal identity.
Negative-angle identity.
Reciprocal identity.
Simplify.
You may start with either side of the given
equation. It is often easier to begin with the more
complicated side and simplify it to match the
simpler side.
Helpful Hint
Check It Out! Example 1a
Prove each trigonometric identity.
sin θ cot θ = cos θ
cos θ = cos θ
Choose the left-hand side
to modify.
Ratio identity.
Simplify.
cos θ
Check It Out! Example 1b
Prove each trigonometric identity.
1 – sec(–θ) = 1 – secθ
Choose the left-hand side
to modify.
Reciprocal identity.
Negative-angle identity.
Reciprocal Identity.
You can use the fundamental trigonometric
identities to simplify expressions.
If you get stuck, try converting all of the
trigonometric functions to sine and cosine
functions.
Helpful Hint
Example 2A: Using Trigonometric Identities to
Rewrite Trigonometric Expressions
Rewrite each expression in terms of cos θ,
and simplify.
sec θ (1 – sin2
θ)
cos θ
Substitute.
Multiply.
Simplify.
Example 2B: Using Trigonometric Identities to
Rewrite Trigonometric Expressions
Rewrite each expression in terms of sin θ, cos θ,
and simplify.
sinθ cosθ(tanθ + cotθ)
sin2
θ + cos2
θ
Substitute.
Multiply.
Simplify.
1 Pythagorean identity.
Check It Out! Example 2a
Rewrite each expression in terms of sin θ, and
simplify.
Pythagorean identity.
Simplify.
Factor the difference of two squares.
Check It Out! Example 2b
Rewrite each expression in terms of sin θ, and
simplify.
cot2
θ
csc2
θ – 1 Pythagorean identity.
Substitute.
Simplify.
Example 3: Physics Application
At what angle will a wooden block on a concrete
incline start to move if the coefficient of friction
is 0.62?
Set the expression for the weight component
equal to the expression for the force of friction.
mg sinθ = μmg cosθ
sinθ = μcosθ
sinθ = 0.62 cosθ
Divide both sides by mg.
Substitute 0.62 for μ.
Example 3 Continued
tanθ = 0.62
θ = 32°
The wooden block will start to move when
the concrete incline is raised to an angle of
about 32°.
Divide both sides by cos θ.
Ratio identity.
Evaluate inverse tangent.
Check It Out! Example 3
Use the equation mg sinθ = μmg cosθ to
determine the angle at which a waxed wood
block on a wood incline with μ = 0.4 begins
to slide.
Set the expression for the weight component
equal to the expression for the force of friction.
mg sinθ = μmg cosθ
sinθ = μcosθ
sinθ = 0.4 cosθ
Divide both sides by mg.
Substitute 0.4 for μ.
Check It Out! Example 3 Continued
tanθ = 0.4
θ = 22°
The wooden block will start to move when
the concrete incline is raised to an angle of
about 22°.
Divide both sides by cos θ.
Ratio identity.
Evaluate inverse tangent.
Lesson Quiz: Part I
1. sinθ secθ =
Prove each trigonometric identity.
2. sec2
θ = 1 + sin2
θ sec2
θ
= 1 + tan2
θ
= sec2
θ
Lesson Quiz: Part II
Rewrite each expression in terms of cos θ,
and simplify.
3. sin2
θ cot2
θ secθ cosθ
4. 2 cosθ
All rights belong to their
respective owners.
Copyright Disclaimer Under
Section 107 of the
Copyright Act 1976,
allowance is made for "fair
use" for purposes such as
criticism, comment, news
reporting, TEACHING,
scholarship, and research.
Fair use is a use permitted
by copyright statute that
might otherwise be
infringing.
Non-profit, EDUCATIONAL
or personal use tips the
balance in favor of fair use

More Related Content

What's hot

Algebra 2 unit 9.6
Algebra 2 unit 9.6Algebra 2 unit 9.6
Algebra 2 unit 9.6
Mark Ryder
 
Calculus Ppt
Calculus PptCalculus Ppt
Calculus Ppt
guest5d26f0c
 
Algebra 2 unit 10.3
Algebra 2 unit 10.3Algebra 2 unit 10.3
Algebra 2 unit 10.3
Mark Ryder
 
7.5 apply the tangent ratio
7.5 apply the tangent ratio7.5 apply the tangent ratio
7.5 apply the tangent ratio
detwilerr
 
Algebra 2 unit 9.4.9.5
Algebra 2 unit 9.4.9.5Algebra 2 unit 9.4.9.5
Algebra 2 unit 9.4.9.5
Mark Ryder
 
Sol24
Sol24Sol24
Sim (brhs)
Sim (brhs)Sim (brhs)
Sim (brhs)
Hilda Dragon
 
Trigonometry functions of general angles reference angles
Trigonometry functions of general angles reference anglesTrigonometry functions of general angles reference angles
Trigonometry functions of general angles reference angles
Jessica Garcia
 
Geometry unit 8.2
Geometry unit 8.2Geometry unit 8.2
Geometry unit 8.2
Mark Ryder
 
Couple 136
Couple 136Couple 136
Couple 136
sibrahim102
 
Geometry unit 12.4 segment lengths
Geometry unit 12.4 segment lengthsGeometry unit 12.4 segment lengths
Geometry unit 12.4 segment lengths
Mark Ryder
 
Trigonometry for class xi
Trigonometry for class xiTrigonometry for class xi
Trigonometry for class xi
indu psthakur
 
Algebra 2 unit 9.1
Algebra 2 unit 9.1Algebra 2 unit 9.1
Algebra 2 unit 9.1
Mark Ryder
 
Geometry unit 8.1
Geometry unit 8.1Geometry unit 8.1
Geometry unit 8.1
Mark Ryder
 
Chapter 5 unit f 001
Chapter 5 unit f 001Chapter 5 unit f 001
Chapter 5 unit f 001
jbianco9910
 
4.3 finding missing angles
4.3 finding missing angles4.3 finding missing angles
4.3 finding missing angles
hisema01
 
Calc 4.2a
Calc 4.2aCalc 4.2a
Calc 4.2a
hartcher
 
Unit 4.2
Unit 4.2Unit 4.2
Unit 4.2
Mark Ryder
 
Grade 11, U1A-L2 Motion Graphs
Grade 11, U1A-L2 Motion GraphsGrade 11, U1A-L2 Motion Graphs
Grade 11, U1A-L2 Motion Graphs
gruszecki1
 
Geometry 201 unit 5.7
Geometry 201 unit 5.7Geometry 201 unit 5.7
Geometry 201 unit 5.7
Mark Ryder
 

What's hot (20)

Algebra 2 unit 9.6
Algebra 2 unit 9.6Algebra 2 unit 9.6
Algebra 2 unit 9.6
 
Calculus Ppt
Calculus PptCalculus Ppt
Calculus Ppt
 
Algebra 2 unit 10.3
Algebra 2 unit 10.3Algebra 2 unit 10.3
Algebra 2 unit 10.3
 
7.5 apply the tangent ratio
7.5 apply the tangent ratio7.5 apply the tangent ratio
7.5 apply the tangent ratio
 
Algebra 2 unit 9.4.9.5
Algebra 2 unit 9.4.9.5Algebra 2 unit 9.4.9.5
Algebra 2 unit 9.4.9.5
 
Sol24
Sol24Sol24
Sol24
 
Sim (brhs)
Sim (brhs)Sim (brhs)
Sim (brhs)
 
Trigonometry functions of general angles reference angles
Trigonometry functions of general angles reference anglesTrigonometry functions of general angles reference angles
Trigonometry functions of general angles reference angles
 
Geometry unit 8.2
Geometry unit 8.2Geometry unit 8.2
Geometry unit 8.2
 
Couple 136
Couple 136Couple 136
Couple 136
 
Geometry unit 12.4 segment lengths
Geometry unit 12.4 segment lengthsGeometry unit 12.4 segment lengths
Geometry unit 12.4 segment lengths
 
Trigonometry for class xi
Trigonometry for class xiTrigonometry for class xi
Trigonometry for class xi
 
Algebra 2 unit 9.1
Algebra 2 unit 9.1Algebra 2 unit 9.1
Algebra 2 unit 9.1
 
Geometry unit 8.1
Geometry unit 8.1Geometry unit 8.1
Geometry unit 8.1
 
Chapter 5 unit f 001
Chapter 5 unit f 001Chapter 5 unit f 001
Chapter 5 unit f 001
 
4.3 finding missing angles
4.3 finding missing angles4.3 finding missing angles
4.3 finding missing angles
 
Calc 4.2a
Calc 4.2aCalc 4.2a
Calc 4.2a
 
Unit 4.2
Unit 4.2Unit 4.2
Unit 4.2
 
Grade 11, U1A-L2 Motion Graphs
Grade 11, U1A-L2 Motion GraphsGrade 11, U1A-L2 Motion Graphs
Grade 11, U1A-L2 Motion Graphs
 
Geometry 201 unit 5.7
Geometry 201 unit 5.7Geometry 201 unit 5.7
Geometry 201 unit 5.7
 

Similar to Algebra 2 unit 10.1

Algebra 2 unit 10.6
Algebra 2 unit 10.6Algebra 2 unit 10.6
Algebra 2 unit 10.6
Mark Ryder
 
Math12 lesson 6
Math12 lesson 6Math12 lesson 6
Math12 lesson 6
KathManarang
 
Geometry 201 unit 2.6
Geometry 201 unit 2.6Geometry 201 unit 2.6
Geometry 201 unit 2.6
Mark Ryder
 
3.2 use parallel lines and transversals
3.2 use parallel lines and transversals3.2 use parallel lines and transversals
3.2 use parallel lines and transversals
detwilerr
 
Larson 4.3
Larson 4.3Larson 4.3
Larson 4.3
Sharon Henry
 
Trigonometric identities
Trigonometric identitiesTrigonometric identities
Trigonometric identities
Irishgel Cabasisi
 
Dif calc10week1
Dif calc10week1Dif calc10week1
Dif calc10week1
Carlos Vázquez
 
Guia Fisica Comipems
Guia Fisica ComipemsGuia Fisica Comipems
Guia Fisica Comipems
Toño Avilix
 
8 4 relationships among the functions
8 4 relationships among the functions8 4 relationships among the functions
8 4 relationships among the functions
hisema01
 
Right triangle trigonometry
Right triangle trigonometryRight triangle trigonometry
Right triangle trigonometry
Ramesh Kumar
 
Analytic trigognometry
Analytic trigognometryAnalytic trigognometry
Analytic trigognometry
Siva Palanisamy
 
Geometry 201 unit 3.3
Geometry 201 unit 3.3Geometry 201 unit 3.3
Geometry 201 unit 3.3
Mark Ryder
 
5 3 solving trig eqns
5 3 solving trig eqns5 3 solving trig eqns
5 3 solving trig eqns
Kamarat Kumanukit
 
Foundations of Trigonometry: Navigating Angles and Ratios with Ease"
Foundations of Trigonometry: Navigating Angles and Ratios with Ease"Foundations of Trigonometry: Navigating Angles and Ratios with Ease"
Foundations of Trigonometry: Navigating Angles and Ratios with Ease"
abhishek2019pandey
 
Trignometry
TrignometryTrignometry
1602 parametric equations
1602 parametric equations1602 parametric equations
1602 parametric equations
Dr Fereidoun Dejahang
 
Mathematics
MathematicsMathematics
Mathematics
Michael Nguyen
 
Ch. 1 lecture b
Ch. 1 lecture bCh. 1 lecture b
Ch. 1 lecture b
Brock Banks
 
Trig ratios
Trig ratiosTrig ratios
Trig ratios
rcruz091462
 
Geometry unit 7.2
Geometry unit 7.2Geometry unit 7.2
Geometry unit 7.2
Mark Ryder
 

Similar to Algebra 2 unit 10.1 (20)

Algebra 2 unit 10.6
Algebra 2 unit 10.6Algebra 2 unit 10.6
Algebra 2 unit 10.6
 
Math12 lesson 6
Math12 lesson 6Math12 lesson 6
Math12 lesson 6
 
Geometry 201 unit 2.6
Geometry 201 unit 2.6Geometry 201 unit 2.6
Geometry 201 unit 2.6
 
3.2 use parallel lines and transversals
3.2 use parallel lines and transversals3.2 use parallel lines and transversals
3.2 use parallel lines and transversals
 
Larson 4.3
Larson 4.3Larson 4.3
Larson 4.3
 
Trigonometric identities
Trigonometric identitiesTrigonometric identities
Trigonometric identities
 
Dif calc10week1
Dif calc10week1Dif calc10week1
Dif calc10week1
 
Guia Fisica Comipems
Guia Fisica ComipemsGuia Fisica Comipems
Guia Fisica Comipems
 
8 4 relationships among the functions
8 4 relationships among the functions8 4 relationships among the functions
8 4 relationships among the functions
 
Right triangle trigonometry
Right triangle trigonometryRight triangle trigonometry
Right triangle trigonometry
 
Analytic trigognometry
Analytic trigognometryAnalytic trigognometry
Analytic trigognometry
 
Geometry 201 unit 3.3
Geometry 201 unit 3.3Geometry 201 unit 3.3
Geometry 201 unit 3.3
 
5 3 solving trig eqns
5 3 solving trig eqns5 3 solving trig eqns
5 3 solving trig eqns
 
Foundations of Trigonometry: Navigating Angles and Ratios with Ease"
Foundations of Trigonometry: Navigating Angles and Ratios with Ease"Foundations of Trigonometry: Navigating Angles and Ratios with Ease"
Foundations of Trigonometry: Navigating Angles and Ratios with Ease"
 
Trignometry
TrignometryTrignometry
Trignometry
 
1602 parametric equations
1602 parametric equations1602 parametric equations
1602 parametric equations
 
Mathematics
MathematicsMathematics
Mathematics
 
Ch. 1 lecture b
Ch. 1 lecture bCh. 1 lecture b
Ch. 1 lecture b
 
Trig ratios
Trig ratiosTrig ratios
Trig ratios
 
Geometry unit 7.2
Geometry unit 7.2Geometry unit 7.2
Geometry unit 7.2
 

More from Mark Ryder

Geometry 201 Unit 4.1
Geometry 201 Unit 4.1Geometry 201 Unit 4.1
Geometry 201 Unit 4.1
Mark Ryder
 
Algebra 302 unit 11.4
Algebra 302 unit 11.4Algebra 302 unit 11.4
Algebra 302 unit 11.4
Mark Ryder
 
Algebra 2 unit 10.5
Algebra 2 unit 10.5Algebra 2 unit 10.5
Algebra 2 unit 10.5
Mark Ryder
 
Algebra 2 unit 10.4
Algebra 2 unit 10.4Algebra 2 unit 10.4
Algebra 2 unit 10.4
Mark Ryder
 
11.1 combination and permutations
11.1 combination and permutations11.1 combination and permutations
11.1 combination and permutations
Mark Ryder
 
Unit 11.3 probability of multiple events
Unit 11.3 probability of multiple eventsUnit 11.3 probability of multiple events
Unit 11.3 probability of multiple events
Mark Ryder
 
Unit 11.2 experimental probability
Unit 11.2 experimental probabilityUnit 11.2 experimental probability
Unit 11.2 experimental probability
Mark Ryder
 
Unit 11.2 theoretical probability
Unit 11.2 theoretical probabilityUnit 11.2 theoretical probability
Unit 11.2 theoretical probability
Mark Ryder
 
11.1 11.1 combination and permutations
11.1 11.1 combination and permutations11.1 11.1 combination and permutations
11.1 11.1 combination and permutations
Mark Ryder
 
Geometry 201 unit 5.5
Geometry 201 unit 5.5Geometry 201 unit 5.5
Geometry 201 unit 5.5
Mark Ryder
 
Geometry 201 unit 5.4
Geometry 201 unit 5.4Geometry 201 unit 5.4
Geometry 201 unit 5.4
Mark Ryder
 
Geometry 201 unit 5.3
Geometry 201 unit 5.3Geometry 201 unit 5.3
Geometry 201 unit 5.3
Mark Ryder
 
Geometry 201 unit 4.7
Geometry 201 unit 4.7Geometry 201 unit 4.7
Geometry 201 unit 4.7
Mark Ryder
 
Geometry 201 unit 4.4
Geometry 201 unit 4.4Geometry 201 unit 4.4
Geometry 201 unit 4.4
Mark Ryder
 
Geometry 201 unit 4.3
Geometry 201 unit 4.3Geometry 201 unit 4.3
Geometry 201 unit 4.3
Mark Ryder
 
Geometry 201 unit 4.2
Geometry 201 unit 4.2Geometry 201 unit 4.2
Geometry 201 unit 4.2
Mark Ryder
 
Geometry 201 unit 3.4
Geometry 201 unit 3.4Geometry 201 unit 3.4
Geometry 201 unit 3.4
Mark Ryder
 
Geometry 201 unit 3.1
Geometry 201 unit 3.1Geometry 201 unit 3.1
Geometry 201 unit 3.1
Mark Ryder
 
Geometry 201 unit 2.5
Geometry 201 unit 2.5Geometry 201 unit 2.5
Geometry 201 unit 2.5
Mark Ryder
 
Geometry 201 unit 2.3
Geometry 201 unit 2.3Geometry 201 unit 2.3
Geometry 201 unit 2.3
Mark Ryder
 

More from Mark Ryder (20)

Geometry 201 Unit 4.1
Geometry 201 Unit 4.1Geometry 201 Unit 4.1
Geometry 201 Unit 4.1
 
Algebra 302 unit 11.4
Algebra 302 unit 11.4Algebra 302 unit 11.4
Algebra 302 unit 11.4
 
Algebra 2 unit 10.5
Algebra 2 unit 10.5Algebra 2 unit 10.5
Algebra 2 unit 10.5
 
Algebra 2 unit 10.4
Algebra 2 unit 10.4Algebra 2 unit 10.4
Algebra 2 unit 10.4
 
11.1 combination and permutations
11.1 combination and permutations11.1 combination and permutations
11.1 combination and permutations
 
Unit 11.3 probability of multiple events
Unit 11.3 probability of multiple eventsUnit 11.3 probability of multiple events
Unit 11.3 probability of multiple events
 
Unit 11.2 experimental probability
Unit 11.2 experimental probabilityUnit 11.2 experimental probability
Unit 11.2 experimental probability
 
Unit 11.2 theoretical probability
Unit 11.2 theoretical probabilityUnit 11.2 theoretical probability
Unit 11.2 theoretical probability
 
11.1 11.1 combination and permutations
11.1 11.1 combination and permutations11.1 11.1 combination and permutations
11.1 11.1 combination and permutations
 
Geometry 201 unit 5.5
Geometry 201 unit 5.5Geometry 201 unit 5.5
Geometry 201 unit 5.5
 
Geometry 201 unit 5.4
Geometry 201 unit 5.4Geometry 201 unit 5.4
Geometry 201 unit 5.4
 
Geometry 201 unit 5.3
Geometry 201 unit 5.3Geometry 201 unit 5.3
Geometry 201 unit 5.3
 
Geometry 201 unit 4.7
Geometry 201 unit 4.7Geometry 201 unit 4.7
Geometry 201 unit 4.7
 
Geometry 201 unit 4.4
Geometry 201 unit 4.4Geometry 201 unit 4.4
Geometry 201 unit 4.4
 
Geometry 201 unit 4.3
Geometry 201 unit 4.3Geometry 201 unit 4.3
Geometry 201 unit 4.3
 
Geometry 201 unit 4.2
Geometry 201 unit 4.2Geometry 201 unit 4.2
Geometry 201 unit 4.2
 
Geometry 201 unit 3.4
Geometry 201 unit 3.4Geometry 201 unit 3.4
Geometry 201 unit 3.4
 
Geometry 201 unit 3.1
Geometry 201 unit 3.1Geometry 201 unit 3.1
Geometry 201 unit 3.1
 
Geometry 201 unit 2.5
Geometry 201 unit 2.5Geometry 201 unit 2.5
Geometry 201 unit 2.5
 
Geometry 201 unit 2.3
Geometry 201 unit 2.3Geometry 201 unit 2.3
Geometry 201 unit 2.3
 

Recently uploaded

Your Skill Boost Masterclass: Strategies for Effective Upskilling
Your Skill Boost Masterclass: Strategies for Effective UpskillingYour Skill Boost Masterclass: Strategies for Effective Upskilling
Your Skill Boost Masterclass: Strategies for Effective Upskilling
Excellence Foundation for South Sudan
 
The History of Stoke Newington Street Names
The History of Stoke Newington Street NamesThe History of Stoke Newington Street Names
The History of Stoke Newington Street Names
History of Stoke Newington
 
Reimagining Your Library Space: How to Increase the Vibes in Your Library No ...
Reimagining Your Library Space: How to Increase the Vibes in Your Library No ...Reimagining Your Library Space: How to Increase the Vibes in Your Library No ...
Reimagining Your Library Space: How to Increase the Vibes in Your Library No ...
Diana Rendina
 
BBR 2024 Summer Sessions Interview Training
BBR  2024 Summer Sessions Interview TrainingBBR  2024 Summer Sessions Interview Training
BBR 2024 Summer Sessions Interview Training
Katrina Pritchard
 
What is Digital Literacy? A guest blog from Andy McLaughlin, University of Ab...
What is Digital Literacy? A guest blog from Andy McLaughlin, University of Ab...What is Digital Literacy? A guest blog from Andy McLaughlin, University of Ab...
What is Digital Literacy? A guest blog from Andy McLaughlin, University of Ab...
GeorgeMilliken2
 
Advanced Java[Extra Concepts, Not Difficult].docx
Advanced Java[Extra Concepts, Not Difficult].docxAdvanced Java[Extra Concepts, Not Difficult].docx
Advanced Java[Extra Concepts, Not Difficult].docx
adhitya5119
 
BÀI TẬP BỔ TRỢ TIẾNG ANH 8 CẢ NĂM - GLOBAL SUCCESS - NĂM HỌC 2023-2024 (CÓ FI...
BÀI TẬP BỔ TRỢ TIẾNG ANH 8 CẢ NĂM - GLOBAL SUCCESS - NĂM HỌC 2023-2024 (CÓ FI...BÀI TẬP BỔ TRỢ TIẾNG ANH 8 CẢ NĂM - GLOBAL SUCCESS - NĂM HỌC 2023-2024 (CÓ FI...
BÀI TẬP BỔ TRỢ TIẾNG ANH 8 CẢ NĂM - GLOBAL SUCCESS - NĂM HỌC 2023-2024 (CÓ FI...
Nguyen Thanh Tu Collection
 
How to Fix the Import Error in the Odoo 17
How to Fix the Import Error in the Odoo 17How to Fix the Import Error in the Odoo 17
How to Fix the Import Error in the Odoo 17
Celine George
 
writing about opinions about Australia the movie
writing about opinions about Australia the moviewriting about opinions about Australia the movie
writing about opinions about Australia the movie
Nicholas Montgomery
 
BÀI TẬP BỔ TRỢ TIẾNG ANH LỚP 9 CẢ NĂM - GLOBAL SUCCESS - NĂM HỌC 2024-2025 - ...
BÀI TẬP BỔ TRỢ TIẾNG ANH LỚP 9 CẢ NĂM - GLOBAL SUCCESS - NĂM HỌC 2024-2025 - ...BÀI TẬP BỔ TRỢ TIẾNG ANH LỚP 9 CẢ NĂM - GLOBAL SUCCESS - NĂM HỌC 2024-2025 - ...
BÀI TẬP BỔ TRỢ TIẾNG ANH LỚP 9 CẢ NĂM - GLOBAL SUCCESS - NĂM HỌC 2024-2025 - ...
Nguyen Thanh Tu Collection
 
Pollock and Snow "DEIA in the Scholarly Landscape, Session One: Setting Expec...
Pollock and Snow "DEIA in the Scholarly Landscape, Session One: Setting Expec...Pollock and Snow "DEIA in the Scholarly Landscape, Session One: Setting Expec...
Pollock and Snow "DEIA in the Scholarly Landscape, Session One: Setting Expec...
National Information Standards Organization (NISO)
 
How to Setup Warehouse & Location in Odoo 17 Inventory
How to Setup Warehouse & Location in Odoo 17 InventoryHow to Setup Warehouse & Location in Odoo 17 Inventory
How to Setup Warehouse & Location in Odoo 17 Inventory
Celine George
 
Wound healing PPT
Wound healing PPTWound healing PPT
Wound healing PPT
Jyoti Chand
 
How to Make a Field Mandatory in Odoo 17
How to Make a Field Mandatory in Odoo 17How to Make a Field Mandatory in Odoo 17
How to Make a Field Mandatory in Odoo 17
Celine George
 
Pengantar Penggunaan Flutter - Dart programming language1.pptx
Pengantar Penggunaan Flutter - Dart programming language1.pptxPengantar Penggunaan Flutter - Dart programming language1.pptx
Pengantar Penggunaan Flutter - Dart programming language1.pptx
Fajar Baskoro
 
Liberal Approach to the Study of Indian Politics.pdf
Liberal Approach to the Study of Indian Politics.pdfLiberal Approach to the Study of Indian Politics.pdf
Liberal Approach to the Study of Indian Politics.pdf
WaniBasim
 
Chapter wise All Notes of First year Basic Civil Engineering.pptx
Chapter wise All Notes of First year Basic Civil Engineering.pptxChapter wise All Notes of First year Basic Civil Engineering.pptx
Chapter wise All Notes of First year Basic Civil Engineering.pptx
Denish Jangid
 
Film vocab for eal 3 students: Australia the movie
Film vocab for eal 3 students: Australia the movieFilm vocab for eal 3 students: Australia the movie
Film vocab for eal 3 students: Australia the movie
Nicholas Montgomery
 
Natural birth techniques - Mrs.Akanksha Trivedi Rama University
Natural birth techniques - Mrs.Akanksha Trivedi Rama UniversityNatural birth techniques - Mrs.Akanksha Trivedi Rama University
Natural birth techniques - Mrs.Akanksha Trivedi Rama University
Akanksha trivedi rama nursing college kanpur.
 
PIMS Job Advertisement 2024.pdf Islamabad
PIMS Job Advertisement 2024.pdf IslamabadPIMS Job Advertisement 2024.pdf Islamabad
PIMS Job Advertisement 2024.pdf Islamabad
AyyanKhan40
 

Recently uploaded (20)

Your Skill Boost Masterclass: Strategies for Effective Upskilling
Your Skill Boost Masterclass: Strategies for Effective UpskillingYour Skill Boost Masterclass: Strategies for Effective Upskilling
Your Skill Boost Masterclass: Strategies for Effective Upskilling
 
The History of Stoke Newington Street Names
The History of Stoke Newington Street NamesThe History of Stoke Newington Street Names
The History of Stoke Newington Street Names
 
Reimagining Your Library Space: How to Increase the Vibes in Your Library No ...
Reimagining Your Library Space: How to Increase the Vibes in Your Library No ...Reimagining Your Library Space: How to Increase the Vibes in Your Library No ...
Reimagining Your Library Space: How to Increase the Vibes in Your Library No ...
 
BBR 2024 Summer Sessions Interview Training
BBR  2024 Summer Sessions Interview TrainingBBR  2024 Summer Sessions Interview Training
BBR 2024 Summer Sessions Interview Training
 
What is Digital Literacy? A guest blog from Andy McLaughlin, University of Ab...
What is Digital Literacy? A guest blog from Andy McLaughlin, University of Ab...What is Digital Literacy? A guest blog from Andy McLaughlin, University of Ab...
What is Digital Literacy? A guest blog from Andy McLaughlin, University of Ab...
 
Advanced Java[Extra Concepts, Not Difficult].docx
Advanced Java[Extra Concepts, Not Difficult].docxAdvanced Java[Extra Concepts, Not Difficult].docx
Advanced Java[Extra Concepts, Not Difficult].docx
 
BÀI TẬP BỔ TRỢ TIẾNG ANH 8 CẢ NĂM - GLOBAL SUCCESS - NĂM HỌC 2023-2024 (CÓ FI...
BÀI TẬP BỔ TRỢ TIẾNG ANH 8 CẢ NĂM - GLOBAL SUCCESS - NĂM HỌC 2023-2024 (CÓ FI...BÀI TẬP BỔ TRỢ TIẾNG ANH 8 CẢ NĂM - GLOBAL SUCCESS - NĂM HỌC 2023-2024 (CÓ FI...
BÀI TẬP BỔ TRỢ TIẾNG ANH 8 CẢ NĂM - GLOBAL SUCCESS - NĂM HỌC 2023-2024 (CÓ FI...
 
How to Fix the Import Error in the Odoo 17
How to Fix the Import Error in the Odoo 17How to Fix the Import Error in the Odoo 17
How to Fix the Import Error in the Odoo 17
 
writing about opinions about Australia the movie
writing about opinions about Australia the moviewriting about opinions about Australia the movie
writing about opinions about Australia the movie
 
BÀI TẬP BỔ TRỢ TIẾNG ANH LỚP 9 CẢ NĂM - GLOBAL SUCCESS - NĂM HỌC 2024-2025 - ...
BÀI TẬP BỔ TRỢ TIẾNG ANH LỚP 9 CẢ NĂM - GLOBAL SUCCESS - NĂM HỌC 2024-2025 - ...BÀI TẬP BỔ TRỢ TIẾNG ANH LỚP 9 CẢ NĂM - GLOBAL SUCCESS - NĂM HỌC 2024-2025 - ...
BÀI TẬP BỔ TRỢ TIẾNG ANH LỚP 9 CẢ NĂM - GLOBAL SUCCESS - NĂM HỌC 2024-2025 - ...
 
Pollock and Snow "DEIA in the Scholarly Landscape, Session One: Setting Expec...
Pollock and Snow "DEIA in the Scholarly Landscape, Session One: Setting Expec...Pollock and Snow "DEIA in the Scholarly Landscape, Session One: Setting Expec...
Pollock and Snow "DEIA in the Scholarly Landscape, Session One: Setting Expec...
 
How to Setup Warehouse & Location in Odoo 17 Inventory
How to Setup Warehouse & Location in Odoo 17 InventoryHow to Setup Warehouse & Location in Odoo 17 Inventory
How to Setup Warehouse & Location in Odoo 17 Inventory
 
Wound healing PPT
Wound healing PPTWound healing PPT
Wound healing PPT
 
How to Make a Field Mandatory in Odoo 17
How to Make a Field Mandatory in Odoo 17How to Make a Field Mandatory in Odoo 17
How to Make a Field Mandatory in Odoo 17
 
Pengantar Penggunaan Flutter - Dart programming language1.pptx
Pengantar Penggunaan Flutter - Dart programming language1.pptxPengantar Penggunaan Flutter - Dart programming language1.pptx
Pengantar Penggunaan Flutter - Dart programming language1.pptx
 
Liberal Approach to the Study of Indian Politics.pdf
Liberal Approach to the Study of Indian Politics.pdfLiberal Approach to the Study of Indian Politics.pdf
Liberal Approach to the Study of Indian Politics.pdf
 
Chapter wise All Notes of First year Basic Civil Engineering.pptx
Chapter wise All Notes of First year Basic Civil Engineering.pptxChapter wise All Notes of First year Basic Civil Engineering.pptx
Chapter wise All Notes of First year Basic Civil Engineering.pptx
 
Film vocab for eal 3 students: Australia the movie
Film vocab for eal 3 students: Australia the movieFilm vocab for eal 3 students: Australia the movie
Film vocab for eal 3 students: Australia the movie
 
Natural birth techniques - Mrs.Akanksha Trivedi Rama University
Natural birth techniques - Mrs.Akanksha Trivedi Rama UniversityNatural birth techniques - Mrs.Akanksha Trivedi Rama University
Natural birth techniques - Mrs.Akanksha Trivedi Rama University
 
PIMS Job Advertisement 2024.pdf Islamabad
PIMS Job Advertisement 2024.pdf IslamabadPIMS Job Advertisement 2024.pdf Islamabad
PIMS Job Advertisement 2024.pdf Islamabad
 

Algebra 2 unit 10.1

  • 1. Holt Algebra 2 UNIT 10.1UNIT 10.1 TRIGONOMETRICTRIGONOMETRIC IDENTITIESIDENTITIES
  • 3. Use fundamental trigonometric identities to simplify and rewrite expressions and to verify other identities. Objective
  • 4. You can use trigonometric identities to simplify trigonometric expressions. Recall that an identity is a mathematical statement that is true for all values of the variables for which the statement is defined.
  • 5. A derivation for a Pythagorean identity is shown below. x2 + y2 = r2 cos2 θ + sin2 θ = 1 Pythagorean Theorem Divide both sides by r2 . Substitute cos θ for and sin θ for
  • 6. To prove that an equation is an identity, alter one side of the equation until it is the same as the other side. Justify your steps by using the fundamental identities.
  • 7. Example 1A: Proving Trigonometric Identities Prove each trigonometric identity. Choose the right-hand side to modify. Reciprocal identities. Simplify. Ratio identity.
  • 8. Example 1B: Proving Trigonometric Identities Prove each trigonometric identity. 1 – cot θ = 1 + cot(–θ) = 1 + (–cotθ) = 1 – cotθ Choose the right-hand side to modify. Reciprocal identity. Negative-angle identity. Reciprocal identity. Simplify.
  • 9. You may start with either side of the given equation. It is often easier to begin with the more complicated side and simplify it to match the simpler side. Helpful Hint
  • 10. Check It Out! Example 1a Prove each trigonometric identity. sin θ cot θ = cos θ cos θ = cos θ Choose the left-hand side to modify. Ratio identity. Simplify. cos θ
  • 11. Check It Out! Example 1b Prove each trigonometric identity. 1 – sec(–θ) = 1 – secθ Choose the left-hand side to modify. Reciprocal identity. Negative-angle identity. Reciprocal Identity.
  • 12. You can use the fundamental trigonometric identities to simplify expressions. If you get stuck, try converting all of the trigonometric functions to sine and cosine functions. Helpful Hint
  • 13. Example 2A: Using Trigonometric Identities to Rewrite Trigonometric Expressions Rewrite each expression in terms of cos θ, and simplify. sec θ (1 – sin2 θ) cos θ Substitute. Multiply. Simplify.
  • 14. Example 2B: Using Trigonometric Identities to Rewrite Trigonometric Expressions Rewrite each expression in terms of sin θ, cos θ, and simplify. sinθ cosθ(tanθ + cotθ) sin2 θ + cos2 θ Substitute. Multiply. Simplify. 1 Pythagorean identity.
  • 15. Check It Out! Example 2a Rewrite each expression in terms of sin θ, and simplify. Pythagorean identity. Simplify. Factor the difference of two squares.
  • 16. Check It Out! Example 2b Rewrite each expression in terms of sin θ, and simplify. cot2 θ csc2 θ – 1 Pythagorean identity. Substitute. Simplify.
  • 17. Example 3: Physics Application At what angle will a wooden block on a concrete incline start to move if the coefficient of friction is 0.62? Set the expression for the weight component equal to the expression for the force of friction. mg sinθ = μmg cosθ sinθ = μcosθ sinθ = 0.62 cosθ Divide both sides by mg. Substitute 0.62 for μ.
  • 18. Example 3 Continued tanθ = 0.62 θ = 32° The wooden block will start to move when the concrete incline is raised to an angle of about 32°. Divide both sides by cos θ. Ratio identity. Evaluate inverse tangent.
  • 19. Check It Out! Example 3 Use the equation mg sinθ = μmg cosθ to determine the angle at which a waxed wood block on a wood incline with μ = 0.4 begins to slide. Set the expression for the weight component equal to the expression for the force of friction. mg sinθ = μmg cosθ sinθ = μcosθ sinθ = 0.4 cosθ Divide both sides by mg. Substitute 0.4 for μ.
  • 20. Check It Out! Example 3 Continued tanθ = 0.4 θ = 22° The wooden block will start to move when the concrete incline is raised to an angle of about 22°. Divide both sides by cos θ. Ratio identity. Evaluate inverse tangent.
  • 21. Lesson Quiz: Part I 1. sinθ secθ = Prove each trigonometric identity. 2. sec2 θ = 1 + sin2 θ sec2 θ = 1 + tan2 θ = sec2 θ
  • 22. Lesson Quiz: Part II Rewrite each expression in terms of cos θ, and simplify. 3. sin2 θ cot2 θ secθ cosθ 4. 2 cosθ
  • 23. All rights belong to their respective owners. Copyright Disclaimer Under Section 107 of the Copyright Act 1976, allowance is made for "fair use" for purposes such as criticism, comment, news reporting, TEACHING, scholarship, and research. Fair use is a use permitted by copyright statute that might otherwise be infringing. Non-profit, EDUCATIONAL or personal use tips the balance in favor of fair use