Chapter 4

Analytic TrigonometrySection 4.3

Double-Angle, Half-Angle and Product-Sum 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.

xx

xxxxxx

cossin2

sincoscossin)sin(

xxx cossin2)2sin(

x

x

x

xx

xx

xxxxxx

2

2

2

22

22

sin21

1)sin1(2

1cos2

)cos1(cos

sincos

sinsincoscos)cos(

x

x

xxx

2

2

22

sin21

1cos2

sincos)2cos(

xx

xxxxxx

2tan1tan2

tantan1tantan)tan(

xxx 2tan1

tan2)2tan(

Example: If x is an angle in quadrant IV with cos x = ⅓ find cos(2x) and sin(2x).

97

922

312 1121cos2)2cos( xx

To apply the double angle formula we need to find sin x first.

38

98

912

312 11cos1sin xx

982

31

382cossin2)2sin( xxx

Now we can apply the double angle formula for sine.

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.

2)2cos(12

2

2

cos

cos2)2cos(1

1cos2)2cos(

xx

xx

xx

2)2cos(12

2

2

sin

)2cos(1sin2

sin21)2cos(

xx

xx

xx

We show how these can be used to reduce the powers on the expression below.

8)4cos()2cos(43

8)4cos(1)2cos(42

4

)2cos(21

4)2(cos)2cos(21

2

2)2cos(1

22

4

2)4cos(1

2

sin

sin

xx

xx

x

xx

x

x

x

x

8)4cos(1

8)4cos(1(2

4

1

4)2(cos1

2)2cos(1

2)2cos(1

22

2)4cos(1

2

cossin

x

x

x

xx

x

xx

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.

2cos1

2sin uu

2cos1

2cos uu

uuu

cos1sin

2tan

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.

222

422

2

1

2225cos1

2225

22

cos

5.112cos

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()sin(cossin

cossin2)sin()sin(

sincoscossin)sin(

sincoscossin)sin(

21 tststs

tststs

tststs

tststs

Product-To-Sum

)cos()cos(sinsin

)cos()cos(coscos

)sin()sin(sincos

)sin()sin(cossin

21

21

21

21

tststs

tststs

tststs

tststs

Sum-To-Product

22

22

22

22

sinsin2coscos

coscos2coscos

sincos2sinsin

cossin2sinsin

tsts

tsts

tsts

tsts

ts

ts

ts

ts

Simplify: cos(4x) sin(7x)

)3sin()11sin(

)3sin()11sin(

)74sin()74sin()7sin()4cos(

21

21

21

xx

xx

xxxxxx