DOUBLE-ANGLE AND HALF-

ANGLE IDENTITIES

If we want to know a formula for we could use the sum formula.

sin 2x

sin 2 sin sin cos cos sinx x x x x x x

we can trade these places

sin cos sin cos 2sin cosx x x x x x This is called the double angle formula for sine since it tells you the sine of double x

sin 2 2sin cosx x x

Let's try the same thing for cos 2x

cos 2 cos cos cos sin sinx x x x x x x 2 2cos sinx x

This is the double angle formula for cosine but by substituting some identities we can express it in a couple other ways.

2 2cos 2 cos sinx x x

2 2cos 1 sinx x

2 21 sin sinx x 21 2sin x 2 2sin 1 cosx x

2 2cos 1 cosx x 22cos 1x

Double-angle Formula for Tangent

2

2 tantan 2

1 tan

xx

x

tan tantan 2 tan

1 tan tan

x xx x x

x x

Summary ofDouble-Angle Formulas

2 2

2

2

sin 2 2sin cos

cos 2 cos sin

cos 2 1 2sin

cos 2 2cos 1

x x x

x x x

x x

x x

2

2 tantan 2

1 tan

xx

x

4sin ,

5 2x x

Find sin 2x

sin 2 2sin cosx x x

xx’4

5

-3

4 3sin 2 2

5 5x

Use triangle to find values.

Let's draw a picture.

25

24

We can also derive formulas for an angle divided by 2 (called the half angle formula). We’ll do this by using the double angle formula for cosine that we found.

Let’s solve this for sin 2cos 2 1 2sin

21 cos 2sin

2

1 cos 2sin

2

Now let = x/2

1 cossin

2 2

x x

In this formula it is NOT both + and - but you must figure out where the terminal side of the angle is and put on the appropriate sign for that quadrant.

We can also derive a half angle formula for cosine in a similar manner. We’ll do this by using a different version of the double angle formula for cosine.

Let’s solve this for cos 2cos 2 2cos 1

21 cos 2cos

2

1 cos 2cos

2

Now let = x/2

1 coscos

2 2

x x

In this formula it is NOT both + and - but you must figure out where the terminal side of the angle is and put on the appropriate sign for that quadrant.

Now to derive a half angle formula for tangent, let’s use the fact that we know that tangent is sine over cosine and use their half angle formulas.

sin2tan

2 cos2

xx

x

1 cos 22

1 cos 22

1 costan

2 1 cos

x x

x

Half-Angle Formulas

1 cossin

2 2

1 coscos

2 2

1 costan

2 1 cos

where the or - is determined by what quadrant is in.2

x x

x x

x x

xx

Summary

As stated it is NOT both + and - but you must figure out where the terminal side of the angle is and put on the appropriate sign for that quadrant.

1 cos sintan

2 sin 1 cos

x x x

x x

You can also derive identities for

the half-angle formulas for tan2

in a couple other forms.

x

We could find sin 15° using the half angle formula.

2

cos1

2sin

Since 15° is half of 30° we could use this formula if x = 30°

30° 30°

15° is in first quadrant and sine is positive there so we want the +

223

115sin

2

32

4

32

122

32

15sin

4sin ,

5 2x x

Find sin2

x

xx’4

5

-3

31

5sin

2 2

x

Use triangle to find cosine value.

If is in quadrant II then half would be in quadrant I where sine is positive

5

52

1 cossin

2 2

x x

5

52

5

2

5

4

1258

253

1