1. 9-3: Other Identities
Objectives:
•Use the double angle and
half angle identities for sine,
cosine & tangent
©2009 Roy L. Gover (www.mrgover.com)
)

2. Analysis
Does sin 90º =2 sin 45°?
Does cos 30°= cos 60°?
1
2

3. Analysis
sin( )
sin cos cos sin
 
   
 

From Lesson 9-2:
Let , then…
  x
 

4. Important Idea
sin2 2sin cos

x x x
Double Angle Identity for
sine:

5. Example
If sin x = 4/5 and x is in
the first quadrant, find the
exact value of sin 2x.
…first, find cos x…

6. Try This
If sin x = 3/5 and x is in the
first quadrant, find the exact
value of sin 2x
24
sin 2
25

x

7. Try This
If sin x = 3/5 and x is in
the second quadrant, find
the exact value of sin 2x.
24
sin 2
25
 
x

8. Analysis
cos( )
cos cos sin sin
 
   
 

From Lesson 9-2:
Let , then…
  x
 

9. Important Idea
2 2
cos2 cos sin
 
x x x
Double Angle Identity for
cosine:

10. Double Angle Identity for cos
Three forms...
1. cos 2x = cos2x - sin2x
2. cos 2x = 2cos2x - 1
3. cos 2x = 1 - 2sin2x
Important Idea

11. Example
If sin x is 1/5 and x is in
the first quadrant, find
the exact value of
cos 2x.

12. Try This
If sin x is 2/5 and x is in
the second quadrant, find
the exact value of
cos 2x.
17
cos2
25

x

13. What would you do to
derive a double angle
identity for tangent?

14. Double Angle Identity for
Tangent:
2
2tan
tan 2
1 tan


x
x
x
Important Idea

15. Try This
If , and is in the
first quadrant, find .
5
cos
5

x x
tan2x
Hint: first find …
tan x
4
tan 2
3
 
x

16. Analysis
2
cos2 2cos 1
 
 
1 cos2
cos
2



 
One of the double angle
identities:
Solve for cos  :

17. Analysis
1 cos2
os
2
c



 
Let 2
2
  
x
x  

18. Analysis
1 cos
cos
2 2

 
x x
which results in:
The half angle identity for
cosine.

19. Important Idea
1 cos
sin
2 2

 
x x
1 cos
cos
2 2

 
x x
1 cos
tan
2 1 cos

 

x x
x
The half angle identities:
See
page
596 of
your
text.

20. Example
Use a half angle identity
to find the exact value of
sin 22°30’
Hint: Multiply the given
angle by 2 then divide by 2

21. Example
Use a half angle identity
to find the exact value of
19
cos
12

Hint: Multiply the given
angle by 2 then divide by 2

22. Try This
Use a half angle identity
to find the exact value of
cos 105°
210 2 3
cos105 cos
2 2
 
   

23. Try This
Use a half angle identity to
find the exact value of
5
5 2 2
4
sin sin
8 2 2

 
 
5
sin
8
