double-angle and half-angle formulas

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

Your Turn: Simplify an Expression

• Simplify cot x cos x + sin x.• Click for answer.

x

xx

sin

coscot

xx

xxx

x

xsin

sin

cossincos

sin

cos 2

xxx

xxcsc

sin

1

sin

sincos 22

Your Turn: Cosine Sum and Difference Identities

)4530cos(75cos

45sin30sin45cos30cos

Find the exact value of cos 75°.

Click for answer.

4

26

2

2

2

1

2

2

2

3

Your Turn: Sine Sum and Difference Identities

• Find the exact value of .• Click for answer.

12

7sin

34sin

12

4

12

3sin

12

7sin

3

sin4

cos3

cos4

sin

4

62

2

3

2

2

2

1

2

2

Your Turn: Double-Angle Identities

• If , find sin 2x given sin x < 0.

• Click for answer.

3

1cos x

Your Turn: Double-Angle Identities

1cossin ,3

1cos 22 xxx

1

3

1sin

22 x

3

22sin

9

8sin 2 xx

9

24

3

1

3

222cossin22sin

xxx

Your Turn: Half-Angle Identities

• Use a half-angle identity to find sin 22.5°.• Click for answer.

2

45cos1

2

45sin5.22sin

2

22

4

22

222

1

Objective: 7-4 Double-Angle and Half-Angle Identities

17

Verifying An Identity Using Double Angle

1cot

1cot

2sin1

2cos

Find using the double angle formulas. (no calculator)

1. sin 420° 2. 3. tan 240°

4. 5. cos 300° 6. tan 630°

€

cos3π

2

€

sin2π

3

Find the exact values of sin 2x, cos 2x, and tan 2x using the double angles formulas

1.

2.

€

≤x ≤3π

2

€

tan x =−1

2

€

sin x =−4

5

€

2

≤ x ≤ π