Download Slide 5.1

5.1
Fundamental
Identities
Copyright © 2011 Pearson, Inc.
What you’ll learn about
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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 - 2
Basic Trigonometric Identities
Reciprocal Identites
1
1
csc 
sec 
sin 
cos
1
cot  
tan 
1
sin  
csc
1
tan  
cot 
1
cos 
sec
Quotient Identites
sin 
cos
tan  
cot 
cos
tan 
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Slide 5.1 - 3
Pythagorean Identities
cos   sin   1
1  tan   sec 
cot   1  csc 
2
2
2
2
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2
2
Slide 5.1 - 4
Example Using Identities
Find sin and cos if tan   3 and cos  0.
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Slide 5.1 - 5
Example Using Identities
Find sin and cos if tan   3 and cos  0.
1  tan 2   sec 2 
To find sin , use tan   3
1  9  sec 2 
and cos  1 / 10.
sin 
tan  
cos
sin   cos tan 
sec   10
cos  1 / 10


sin   1 / 10 3
sin   3 / 10
Therefore, cos  1 / 10 and sin  3 / 10
Copyright © 2011 Pearson, Inc.
Slide 5.1 - 6
Cofunction Identities
y
Angle A: sin A 
r
x
cos A 
r
x
Angle B: sin B 
r
y
cos B 
r
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y
tan A 
x
x
cot A 
y
x
tan B 
y
y
cot B 
x
r
sec A 
x
r
csc A 
y
r
sec B 
y
r
csc B 
x
Slide 5.1 - 7
Cofunction Identities


sin      cos 
2



tan      cot 
2



sec      csc 
2

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

cos      sin 
2



cot      tan 
2



csc      sec 
2

Slide 5.1 - 8
Even-Odd Identities
sin(x)   sin x
csc(x)   csc x
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cos(x)  cos x
sec(x)  sec x
tan(x)   tan x
cot(x)   cot x
Slide 5.1 - 9
Example Simplifying by Factoring
and Using Identities
Simplify the expression cos 3 x  cos xsin2 x.
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Slide 5.1 - 10
Example Simplifying by Factoring
and Using Identities
Simplify the expression cos 3 x  cos xsin2 x.
cos 3 x  cos x sin 2 x  cos x(cos2 x  sin 2 x)
 cos x(1)
Pythagorean Identity
 cos x
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Slide 5.1 - 11
Example Simplifying by Expanding
and Using Identities
csc x -1csc x  1

Simplify the expression:
cos2 x
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Slide 5.1 - 12
Example Simplifying by Expanding
and Using Identities
csc x  1csc x  1  csc 2 x  1
cos 2 x
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(a  b)(a  b)  a 2  b 2
cos 2 x
cot 2 x

Pythagorean Identity
2
cos x
cos 2 x
1
cos


cot  
2
2
sin x cos x
sin 
1

sin 2 x
 csc 2 x
Slide 5.1 - 13
Example Solving a Trigonometric
Equation
Find all values of x in the interval 0,2 
sin 3 x
that solve
 tan x.
cos x
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Slide 5.1 - 14
Example Solving a Trigonometric
Equation
3
sin x
 tan x
cos x
sin 3 x sin x

cos x cos x
3
sin x  sin x
Reject the posibility that cos 2 x  0
because it would make both
sides of the original equation
undefined. sin x  0 in the interval
0  x  2 when x  0 and x   .
sin 3 x  sin x  0
sin x(sin x  1)  0
2


sin x  0
or
2
sin
x
cos
x 0
 
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cos 2 x  0
Slide 5.1 - 15