Download Do 7.4

Chapter 7
Section 7.4
Double-Angle and Half-Angle Identities
The angle identities we studied in the last section will generate other identities concerning the angle that are
extremely useful in trigonometry. In particular we will see later when it comes to solving trigonometric equations.
In the sum of angles formulas we had before if we replace the s and t each by an x we get each of the double angle
identities below.
sin( x  x)  sin x cos x  cos x sin x
 2 sin x cos x
sin( 2 x)  2 sin x cos x
cos( x  x)  cos x cos x  sin x sin x
cos( 2 x)  cos x  sin x
 cos 2 x  sin 2 x
 cos 2 x  (1  cos 2 x)
 2 cos 2 x  1
 2(1  sin 2 x)  1
 1  2 sin x
2
x  tan x
tan( x  x)  1tan
 tan x tan x

2 tan x
1 tan 2 x
2
2
 2 cos x  1
2
 1  2 sin x
2
tan( 2x) 
2 tan x
1 tan2 x
Example: If x is an angle in quadrant IV with cos x = ⅓ find cos(2x) and sin(2x).
cos(2 x)  2 cos x 1  2
2

1 2
3
1  1 
2
9
7
9
To apply the double angle formula we need to find sin x first. This will be negative since the
angle is in quadrant IV.
sin x   1  cos x   1  
2

1 2
3
Now we can apply the double angle formula for sine.
  1  
1
9
8
9
   
sin( 2 x)  2 sin x cos x  2
 8
3
1
3

 8
3
2 8
9
These identities can be extended to triple angle (or larger) by applying the sum of angles
identity to the angle 3x=2x+x. These identities are not needed very often but it is possible to
derive them when you need them.
Identities for Lowering Powers
The double angle identities for the
cos(2x) can be rewritten in a different
form to produce identities that reduce
the power on either the sine or cosine.
Again these are very useful when it
comes to solving trigonometric
equations.
cos( 2 x)  1  2 sin x
2
2 sin 2 x  1  cos( 2 x)
sin x 
2
1 cos(2 x )
2
cos( 2 x)  2 cos x  1
2
1  cos( 2 x)  2 cos x
2
cos x 
2
1 cos(2 x )
2
We show how these can be used to reduce the powers on
the expression below.
4
sin x


 sin x
2

2
2
1 cos(2 x )
2

sin x cos x
2

1 cos(2 x ) 2
2

1 2 cos(2 x )  cos 2 ( 2 x )
4

4 x)
1 2 cos(2 x )  1cos(
2
4

2  4 cos(2 x ) 1 cos(4 x )
8

3 4 cos(2 x )  cos(4 x )
8



1 cos 2 ( 2 x )
4

4x)
1 1cos(
2
4

2  (1 cos(4 x )
8

1 cos(4 x )
8
1 cos(2 x )
2

Half-Angle Formulas
The following identities can be
obtained by replacing x by u/2 in
the double angle formulas. The
choice of + or – sign in the sine
and cosine formulas depend on
the quadrant the angle u/2 is in.
sin  
u
2
cos  
u
2
tan 
u
2
1 cos u
2
1 cos u
2
sinu
1 cos u
Find the exact value of cos(112.5), sin(112.5), and
tan(112.5)
Since 112.5 is half of 225 we will use the half-angle
formula for the cos(225/2). Since 112.5 is in the second
quadrant the cosine will be negative and sine positive.
cos 112.5
 cos

225
2

1 cos 225
2
sin 112.5
 sin
225
2

1 cos 225
2

1  2 2
2

1  2 2
2

2 2
4

2 2
4

 2 2
2

2 2
2
tan 112.5°
=
sin 225°
1+cos 225°
=
2
2
2
1−
2
=
− 2
2− 2
−
sin( s  t )  sin s cos t  cos s sin t
sin( s  t )  sin s cos t  cos s sin t
Product-To-Sum and Sum-To-Product Identities:
If we take the sum and difference identities and add
or subtract them we get ways to turn products into
sums or sums into products
Product-To-Sum
sin s cos t  12 sin( s  t )  sin( s  t ) 
sin( s  t )  sin( s  t )
cos s cos t  12 cos( s  t )  cos( s  t ) 
sin s sin t  12 cos( s  t )  cos( s  t ) 
cos s sin t 
1
2
sin( s  t )  sin( s  t )  2 sin s cos t
sin s cos t 
1
2
sin( s  t )  sin( s  t ) 
Sum-To-Product
sin s  sin t  2 sin  s 2t cos s2t 
sin s  sin t  2 cos s 2t sin  s2t 
cos s  cos t  2 cos s 2t cos s2t 
cos s  cos t  2 sin  s 2t sin  s2t 
cos( 4 x) sin( 7 x)  12 sin( 4 x  7 x)  sin( 4 x  7 x) 
Simplify: cos(4x) sin(7x)
 12 sin( 11x)  sin( 3x) 
 12 sin( 11x)  sin( 3x) 