Download Solving Trigonometric Equations by Factoring

Copyright © 2011 Pearson, Inc.
Goal: Use the fundamental identities to simplify trigonometric expressions.
5.1
Fundamental
Identities
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What you’ll learn about
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
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Identities
Basic Trigonometric Identities
Pythagorean Identities
Cofunction Identities
Odd-Even Identities
Simplifying Trigonometric Expressions
Solving Trigonometric Equations
… and why
Identities are important when working with trigonometric
functions in calculus.
Copyright © 2011 Pearson, Inc.
Slide 5.1 - 3
FYI
Copyright © 2011 Pearson, Inc.
Basic Trigonometric Identities
Reciprocal Identites
1
csc 
sin 
1
sec 
cos
1
cot  
tan 
1
sin  
csc
1
cos 
sec
1
tan  
cot 
Quotient Identites
sin 
tan  
cos
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cos
cot 
tan 
Slide 5.1 - 5
Pythagorean Identities
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Slide 5.1 - 6
Cofunction Identities


sin      cos 
2



cos      sin 
2



tan      cot 
2



cot      tan 
2



sec      csc 
2



csc      sec 
2

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Slide 5.1 - 7
Even-Odd Identities
sin(x)   sin x
cos(x)  cos x
tan(x)   tan x
csc(x)   csc x
sec(x)  sec x
cot(x)   cot x
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Slide 5.1 - 8
Simplification Ideas

Rewrite tan, cot, sec, and csc in terms of sin and cos.

Expand Products
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Example: (1 + sin x)(1 – sin x) = 1 – sin x + sin x – sin2x
Factor

Example: sinx•cosx – sinx = sinx(cosx – 1)

Take Square Roots

Combining Fractions
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Simplify by Rewriting
tan 𝑥 ∙ cos 𝑥
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Slide 5.1 - 10
Simplify by Expanding Products
sin 𝑥 tan 𝑥 + cot 𝑥
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Slide 5.1 - 11
Simplifying by Factoring
Simplify the expression cos3 x  cos xsin2 x.
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Slide 5.1 - 12
Simplify by Taking Square Roots
1 + tan2 𝜃
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Slide 5.1 - 13
Simplify by Combining Fractions
sec 𝑥 sin 𝑥
−
sin 𝑥 cos 𝑥
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Slide 5.1 - 14
Example Simplifying by Expanding
and Using Identities
csc x -1csc x  1

Simplify the expression:
cos 2 x
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Slide 5.1 - 15
Goal: Solve trigonometric equations.
5.1
Day 2
Fundamental
Identities
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Warm Up

Simplify.
1 − sin2 𝜃
cos 𝜃
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Slide 5.1 - 17
Example 1: Using the Pythagorean
Identity

Given sin 𝜃 =
exactly.
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3
,
4
use the Pythagorean Identity to find the cos 𝜃
Slide 5.1 - 18
Example 2a: Solving Trigonometric
Equations by Factoring

Find all solutions to the trigonometric equation below.
2 sin2 𝑥 + sin 𝑥 − 1 = 0
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Slide 5.1 - 19
Example 2b: Solving Trigonometric
Equations by Factoring

Find all solutions to the trigonometric equation below.
2 sin2 𝑥 + 3sin 𝑥 = 2
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Slide 5.1 - 20
Example 2c: Solving Trigonometric
Equations by Factoring

Find all solutions to the trigonometric equation in the interval
[0, 2π).
tan 𝑥 sin2 𝑥 = tan 𝑥
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Slide 5.1 - 21