DOUBLE-ANGLE AND HALF-

ANGLE FORMULAS

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

2sin

sincoscossinsin2sin

we can trade these places

cossin2cossincossin This is called the double angle formula for sine since it tells you the sine of double

cossin22sin

Let's try the same thing for 2cos

sinsincoscoscos2cos

22 sincos

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

22 sincos2cos

22 sin1cos

22 sinsin1 2sin21 22 cos1sin

22 cos1cos 1cos2 2

Double-angle Formula for Tangent

tantan

tan2

2

1 2

tantan1

tantantan2tan

Summary ofDouble-Angle Formulas

sin sin cos

cos cos sin

cos sin

cos cos

2 2

2

2 1 2

2 2 1

2 2

2

2

tantan

tan2

2

1 2

Half-Angle Formulas

in. is 2

quadrant by what determined is -or thewhere

cos1

cos1

2tan

2

cos1

2cos

2

cos1

2sin

We can also derive formulas for an angle divided by 2.

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.

cos1

sin

sin

cos1

2tan

2tanfor Formulas Angle-Half

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 = 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

2

,5

4sin

2sin Find

cossin22sin

45

-3

5

3

5

422sin Use triangle to

find values.

Let's draw a picture.

25

24

2

,5

4sin

2sin Find

45

-3

253

1

2sin

Use triangle to find cosine value.

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

5

52

2

cos1

2sin

5

52

5

2

5

4

1258

253

1

Acknowledgement

I wish to thank Shawna Haider from Salt Lake Community College, Utah USA for her hard work in creating this PowerPoint.

www.slcc.edu

Shawna has kindly given permission for this resource to be downloaded from www.mathxtc.com and for it to be modified to suit the Western Australian Mathematics Curriculum.

Stephen CorcoranHead of MathematicsSt Stephen’s School – Carramarwww.ststephens.wa.edu.au