sin2 2sin cos u u u = cos 2 cos sin 2cos 1 1 2sin u u u u u = - = - =

5.5 Multiple-Angle and Product-to-Sum Formulas
Double-Angle Formulas
sin 2u  2sin u cos u
tan 2u 
2 tan u
1  tan 2 u
cos 2u  cos 2 u  sin 2 u
 2 cos 2 u  1
 1  2sin 2 u
Example: Solve 2cos x  sin 2 x  0
Example: If cos  
5
, with theta in the fourth quadrant, find sin 2 ,cos 2 , and tan 2 .
13
Power-Reducing Formulas
sin 2 u 
1  cos 2u
2
cos 2 u 
1  cos 2u
2
tan 2 u 
1  cos 2u
1  cos 2u
Half-Angle Formulas
sin
u
1  cos u

2
2
cos
u
1  cos u

2
2
tan
The signs of the sine and cosine depend on the quadrant that u/2 lies in.
Example: Find the exact value of sin105 .
Example: Find all solutions of 2  sin 2 x  2cos 2
x
in the interval [0, 2 ) .
2
u 1  cos u
sin u


2
sin u
1  cos u
Product-to-Sum Formulas
sin u sin v 
1
cos  u  v   cos  u  v  
2
cos u cos v 
1
cos  u  v   cos  u  v 
2
sin u cos v 
1
sin  u  v   sin  u  v  
2
cos u sin v 
1
sin  u  v   sin  u  v  
2
Sum-to-Product Formulas
uv
u v 
sin u  sin v  2sin 
 cos 

 2 
 2 
u v  u v 
sin u  sin v  2cos 
 sin 

 2   2 
uv
u v 
cos u  cos v  2cos 
 cos 

 2 
 2 
u v  u v 
cos u  cos v  2sin 
 sin 

 2   2 
Example: Find the exact value of cos195  cos105 .