Download Section 7.2

CHAPTER 7:
Trigonometric Identities, Inverse
Functions, and Equations
7.1 Identities: Pythagorean and Sum and Difference
7.2 Identities: Cofunction, Double-Angle, and HalfAngle
7.3 Proving Trigonometric Identities
7.4 Inverses of the Trigonometric Functions
7.5 Solving Trigonometric Equations
Copyright © 2009 Pearson Education, Inc.
7.2
Identities: Cofunction, Double-Angle,
Half-Angle




Use cofunction identities to derive other identities.
Use the double-angle identities to find function values
of twice an angle when one function value is known
for that angle.
Use the half-angle identities to find function values of
half an angle when one function value is known for
that angle.
Simplify trigonometric expressions using the doubleangle identities and the half-angle identities.
Copyright © 2009 Pearson Education, Inc.
Cofunction Identities


sin   x   cos x
2



cos   x   sin x
2



tan   x   cot x
2



cot   x   tan x
2



sec   x   csc x
2



csc   x   sec x
2

Copyright © 2009 Pearson Education, Inc.
Slide 7.2 - 4
Example


Prove the identity sin  x    cos x.

2
Solution:




sin  x    sin x cos  cos x sin

2
2
2
 sin x  0  cos x 1
 cos x
Copyright © 2009 Pearson Education, Inc.
Slide 7.2 - 5
Cofunction Identities


sin  x    cos x

2


sin  x     cos x

2


cos  x     sin x

2


cos  x    sin x

2
Copyright © 2009 Pearson Education, Inc.
Slide 7.2 - 6
Example
Find an identity for each of the following.


a) tan  x  
b) sec x  90º 

2
Solution:


sin  x  



cos
x


2
a) tan  x   

  cot x


2

 sin x

cos  x  

2
b) sec x  90º  
Copyright © 2009 Pearson Education, Inc.
1
1
 csc x

cos x  90º  sin x
Slide 7.2 - 7
Double-Angle Identities
sin 2x  2sin x cos x
cos 2x  cos 2 x  sin 2 x
 1  2sin 2 x
 2 cos 2 x  1
2 tan x
tan 2x 
1  tan 2 x
Copyright © 2009 Pearson Education, Inc.
Slide 7.2 - 8
Other Useful Identities
1  cos 2x
sin x 
2
2
1  cos 2x
cos x 
2
2
1  cos 2x
tan x 
1  cos 2x
2
Copyright © 2009 Pearson Education, Inc.
Slide 7.2 - 9
Example
Find an equivalent expression for each of the following.
a) sec 3 in terms of function values of 
3
b) cos x in terms of function values of x or 2x
raised only to the first power
Solution:
a) sin 3  sin 2     sin 2 cos  cos 2 sin 


 2sin  cos cos  2 cos 2   1 sin 
 2sin cos   2sin cos   sin
2
2
 4 sin cos2   sin
Copyright © 2009 Pearson Education, Inc.
Slide 7.2 - 10
Example
Solution continued:
b) cos3 x
 cos2 x cos x
1  cos 2x

cos x
2
cos x  cos x cos 2x

2
Copyright © 2009 Pearson Education, Inc.
Slide 7.2 - 11
Half-Angle Identities
x
1  cos x
sin  
2
2
x
1 cos x
cos  
2
2
Copyright © 2009 Pearson Education, Inc.
x
1  cos 2x
tan  
2
1  cos 2x
sin x

1  cos x
1  cos x

sin x
Slide 7.2 - 12
Example
Find tan (π/8) exactly. Then check the answer using a
graphing calculator in RADIAN mode.
Solution:
tan


sin

2
2
2
2

2 2
2
4 
 tan 4 

8
2
2
1  cos
1
4
2
2
2 2 2
 2 1



2 2
2 2 2 2
Copyright © 2009 Pearson Education, Inc.
Slide 7.2 - 13
Example
Simplify each of the following.
sin x cos x
2 x
b) 2sin  cos x
a)
1
2
cos 2x
2
Solution:
sin 2x
sin x cos x 2 sin x cos x 2sin x cos x
 


a)
1
2 1 cos 2x
cos 2x
cos 2x
cos 2x
2
2
 tan2x
 1  cos x 
2 x
b) 2sin  cos x  2 
 cos x


2
2 
 1 cos x  cos x
1
Copyright © 2009 Pearson Education, Inc.
Slide 7.2 - 14