Download tan 2θ =

Double and Half Angle Formulas
** More trigonometric formulas - based on the addition formulas from previous section (Gotta Love Them).
Recall:
sin  x  y  
cos  x  y  
tan  x  y  
sin  2  
cos  2  
tan  2  
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Double Formulas
Example: If sin   
3 
,
    , find the exact value of sin  2  and cos  2 
5 2
Example: Find an expression for the triple angle formula
cos  3x 
Example: Find the exact value of the following expression:
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in terms of cosine only.

 5 
sin  2 cos 1   
 13  

Double Angle Formulas
As a consequence of the double angle formulas from cos  2  , we can construct the
following useful identities.
sin 2   
cos 2   
tan 2   
Example: Simplify the following using the double angle formulas until the powers of
sine and cosine are not greater than 1:
sin 2  2 x  cos 2  2 x 
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Half Angle Formulas
As a consequence of the double angle formulas on the previous page, we can construct the half-angle formulas:
1
1 
cos  u   
1  cos  u  
2
2


1
1 
sin  u    1  cos  u  
2
2 
sin  u 
 1  1  cos  u 
tan  u  

sin  u 
1  cos  u 
2 
The +/- depends on the quadrant in which
1
u lies.
2
Example: Find the exact value of
(a)
cos 15

(b)
Example: Find the exact value of the expression:
1
 3 
cos 2  sin 1   
 5 
2
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 7 
csc 

 8 
Double and Half Angle Formulas
Establish the identity:
cot    tan  
 cos  2 
cot    tan  
Example: If x  8 tan   , express
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sin  2 
as a function of x.