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Chapter 5
Trigonometric Identities
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5.1
Fundamental Identities
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Fundamental Identities

Reciprocal Identities
1
cot  
tan 

1
sec 
cos
1
csc 
sin 
Quotient Identities
sin 
tan  
cos
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cos
cot  
sin 
Slide 5-4
More Identities

Pythagorean Identities
sin 2   cos 2   1
tan 2   1  sec 2 
1  cot 2   csc 2 

Negative-Angle Identities
sin( )   sin 
csc( )   csc
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cos( )  cos
sec( )  sec
tan( )   tan 
cot( )   cot 
Slide 5-5
5
tan   
3
Example: If
and  is in quadrant II,
find each function value.
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Slide 5-6
Example: Express One Function in
Terms of Another

Express cot x in
terms of sin x.
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Slide 5-7
Example: Rewriting an Expression in
Terms of Sine and Cosine

Rewrite cot   tan  in terms of sin  and cos  .
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Slide 5-8
66
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Slide 5-9
5.2
Verifying Trigonometric
Identities
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6, 16, 20
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Slide 5-11
Hints for Verifying Identities

1. Learn the fundamental identities given in the last
section. Whenever you see either side of a
fundamental identity, the other side should come to
mind. Also, be aware of equivalent forms of the
2
2
fundamental identities. For example sin   1  cos 
is an alternative form of the identity sin 2   cos 2   1.

2. Try to rewrite the more complicated side of the
equation so that it is identical to the simpler side.
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Slide 5-12
Hints for Verifying Identities continued


3. It is sometimes helpful to express all
trigonometric functions in the equation in terms
of sine and cosine and then simplify the result.
4. Usually, any factoring or indicated algebraic
operations should be performed. For example,
the expression sin 2 x  2sin x  1 can be factored as
(sin x  1) 2 . The sum or difference of two
1
1
trigonometric expressions such as sin   cos , can
be added or subtracted in the same way as any
other rational expression.
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Slide 5-13
Hints for Verifying Identities continued

5. As you select substitutions, keep in mind the
side you are changing, because it represents
your goal. For example, to verify the identity
1
tan x  1 
cos 2 x
2
try to think of an identity that relates tan x to
1
sec
x

cos x. In this case, since
and
cos x
sec 2 x  tan 2 x  1, the secant function is the best
link between the two sides.
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Slide 5-14
Hints for Verifying Identities continued

6. If an expression contains 1 + sin x,
multiplying both the numerator and denominator
by 1  sin x would give 1  sin2 x, which could be
replaced with cos2x. Similar results for 1  sin x,
1 + cos x, and 1  cos x may be useful.

Remember that verifying identities is NOT the
same as solving equations.
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Slide 5-15
Example: Working with One Side

Prove the identity (tan 2 x  1)(cos2 x  1)   tan 2 x
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Slide 5-16
Example: Working with One Side

Prove the identity
1
 csc x  sin x
sec x tan x
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Slide 5-17
Example: Working with One Side

Prove the identity tan x  cot y  tan y  cot x
tan x cot y
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Slide 5-18
Example: Working with Both Sides

Verify that the following equation is an identity.
sec   tan  1  2sin   sin 2 

sec   tan 
cos 2 
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Slide 5-19
5.3
Sum and Difference Identities
for Cosine
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Cosine of a Sum or Difference


cos( A  B)  cos A cos B  sin A sin B
cos( A  B)  cos A cos B  sin A sin B
Find the exact value of cos 75.

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Slide 5-21
More Examples

cos
5
12
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
cos87 cos93  sin 87 sin 93
Slide 5-22
Cofunction Identities


cos(90   )  sin 
cot(90   )  tan 
sin(90   )  cos
sec(90   )  csc
tan(90   )  cot 
csc(90   )  sec
Similar identities can be obtained for a real
number domain by replacing 90 with /2.
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Slide 5-23
Example: Using Cofunction Identities


Find an angle that satisfies sin (20) = cos 
38
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Slide 5-24
Example: Reducing

Write cos (270  ) as a trigonometric function
of .
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Slide 5-25
52
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Slide 5-26
5.4
Sum and Difference Identities
for Sine and Tangent
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Sine of a Sum of Difference


sin( A  B)  sin A cos B  cos A sin B
sin( A  B)  sin A cos B  cos A sin B
Tangent of a Sum or Difference
tan A  tan B
tan( A  B) 
1  tan A tan B
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tan A  tan B
tan( A  B) 
1  tan A tan B
Slide 5-28
Example: Finding Exact Values

Find an exact value for sin 105.
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Slide 5-29
Example: Finding Exact Values
continued

Find an exact value for
sin 90 cos 135  cos 90 sin 135
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Slide 5-30
Example: Write each function as an
expression involving functions of .

sin (30 + )
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
tan (45 + )
Slide 5-31
Example: Finding Function Values and
the Quadrant of A + B

Suppose that A and B are angles in standard
position, with sin A = 4/5, /2 < A < , and cos B
= 5/13,  < B < 3/2. Find sin (A + B), tan (A +
B), and the quadrant of A + B.
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Slide 5-32
53-55
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Slide 5-33
5.5
Double-Angle Identities
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Double-Angle Identities

cos 2 A  cos A  sin A
2
2
cos 2 A  1  2sin A
2
cos 2 A  2cos A  1
2
sin 2 A  2sin A cos A
2 tan A
tan 2 A 
1  tan 2 A
Try 6, 26
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Slide 5-35
Example: Given tan  = 3/5 and sin  < 0,
find sin 2, and cos 2. Problems like 7-15

Find the value of sin.
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Slide 5-36
Example: Multiple-Angle Identity

Find an equivalent expression for cos 4x.
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Slide 5-37
38,54
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Slide 5-38
Product-to-Sum Identities

1
cos A cos B   cos( A  B )  cos( A  B 
2
1
sin A sin B  cos( A  B)  cos( A  B) 
2
1
sin A cos B  sin( A  B)  sin( A  B) 
2
1
cos A sin B  sin( A  B)  sin( A  B) 
2
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Slide 5-39
Example


Write 3sin 2 cos as the sum or difference of
two functions.
1
sin 2 cos  sin(2   )  sin(2   ) 
2
1
 sin(3 )  sin  
2
1
1
 sin 3  sin 
2
2
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Slide 5-40
Sum-to-Product Identities

 A B
 A B
sin A  sin B  2sin 
 cos 

 2 
 2 
 A B  A B
sin A  sin B  2cos 
 sin 

 2   2 
 A B
 A B
cos A  cos B  2cos 
 cos 

 2 
 2 
 A B  A B
cos A  cos B  2sin 
 sin 

 2   2 
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Slide 5-41
5.6
Half-Angle Identities
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Half-Angle Identities

A
1  cos A
cos  
2
2
A
1  cos A
sin  
2
2
A
1  cos A
tan  
2
1  cos A
A 1  cos A
tan 
2
sin A
A
sin A
tan 
2 1  cos A
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Slide 5-43
Example: Finding an Exact Value

Use a half-angle identity to find the exact value
of sin (/8).
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Slide 5-44
Example: Finding an Exact Value

Find the exact value of tan 22.5
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Slide 5-45
24, 30, 40, 54
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Slide 5-46