6.3 Double-Angle & Half-Angle Identities

The Double-Angle Identities can be easily derived from the sum identities

• •

OR

OR

•

2 2

2

2

2

sin( ) sin cos cos sin

2sin cos

cos( ) cos cos sin sin

cos sin

2cos 1

1 2sin

tan tan 2 tantan( )

1 tan tan 1 tan

2 2

2 2

(since sin 1 cos )

(since cos 1 sin )

1 cos 1 cossin tan , cos 1

2 2 2 1 cos1 cos

, sin 0sin

1 cos sincos , cos 1

2 2 1 cos

g g

g

Double Angle Identities

Half-Angle Identities

2 2

2

22

sin 2 2sin cos cos 2 cos sin

2cos 1

2 tantan 2 , tan 1 1 2sin

1 tan

g g

g

Examples: Find the exact values

Ex 1) Find

76

32

7sin

121 7

sin2 6

1 cos

2

21

22

2 3

4

2 3

2

QII so (+)

Ex 2) Given

2 2

2 2

cos 2 cos sin

5 12 25 144 119

13 13 169 169 169

12 3sin and , find cos 2

13 2

–5

13

θ–12

Ex 3) Given3 3

sin and 2 , find cos5 2 2

945 51 cos 1 9 3 3 10

cos2 2 2 2 10 1010

–3

4

5

32

23

4 2

QII so cos is (–)

Is it (+) or (–)?

Ex 4) Given sec 2 and tan 0, find sin2

1 12 21 cos 1 1 1

sin2 2 2 2 4 2

1

2

32

23

4 2

sec (recip of cos) is (+) in QI & QIVtan is (–) in QII & QIV

3

QII so sin is (+)

The horizontal distance a projectile can travel can be found using 2 sin 2v

xg

where v = initial velocityθ = launch angleg = gravitational constant = 9.8 m/s2

Ex 5) A punter consistently kicks the football at a 42° angle with an initial velocity of 25 m/s. How far from the punter is the ball when it hits the ground?

2(25) sin 2(42 ) 625sin8463.4 m

9.8 9.8x

Homework

#602 Pg 302 #1–19 odd, 24, 26, 27, 33, 41, 45, 47, 55