Download Sec 4.3

Chapter 4
Analytic Trigonometry
Section 4.3
Double-Angle, Half-Angle and ProductSum Formulas
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 x  1
2
 2(1  sin 2 x)  1
 1  2 sin 2 x
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
2
1 tan 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  92 1  97
To apply the double angle formula we need to find sin x first.
sin x   1  cos x   1  
2

1 2
3
  1  19  
8
9

 8
3
Now we can apply the double angle formula for sine.
   
sin( 2 x)  2 sin x cos x  2
 8
3
1
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 2 x  1cos(2 2 x )
cos( 2 x)  2 cos 2 x  1
1  cos( 2 x)  2 cos x
2
cos 2 x  1cos(2 2 x )
We show how these can be used to reduce
the powers on the expression below.
sin 2 x cos 2 x
4
sin x


 sin x
2

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(2 x )
2
4 x)
 1cos(
8

1 cos(2 x )
2

Half-Angle Formulas
The following
identities can 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 u2  
1 cos u
2
cos u2  
1 cos u
2
tan u2  1sincosuu
Find the exact value of cos(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.
cos 112.5
 cos
225
2

1 cos 225
2

1  2 2
2

2 2
4

 2 2
2
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
sin( s  t )  sin s cos t  cos s sin t
sin( s  t )  sin s cos t  cos s sin t
sin( s  t )  sin( s  t )  2 sin s cos t
sin s cos t 
1
2
sin( s  t )  sin( s  t ) 
Product-To-Sum
Sum-To-Product
sin( s  t )  sin( s  t )
cos s sin t  12 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 ) 
sin s  sin t  2 sin  s 2t cos s2t 
sin s cos t 
1
2
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) 