Aim What are the double angle and half

Aim: What are the double angle and half angle trig identities? Do Now: Write the identities of the following: 1. sin (A + A) 2. cos (A + A) HW: p. 507 # 12, 14, 18 p. 511 # 10, 12, 14

sin 2 A = sin A cos A+ cos A sin A = sin A cos A + sin A cos A = 2 sin A cos 2 A = cos A – sin A = therefore, cos 2 A = therefore,

To find the identity of tan 2 A, treat 2 A as A + A

Double angle identities • sin 2 A = 2 sin A cos A • cos 2 A = cos 2 A – sin 2 A • cos 2 A = 2 cos 2 A – 1 • cos 2 A = 1 – 2 sin 2 A

Half angle identities How do we determine when to use positive or negative ratio? It depends on the quadrant where the half angle located.

If 0 ≤ θ ≤ 90, then 0 ≤ ½ θ ≤ 45, therefore, ½ θ is on quadrant I If 90 ≤ θ ≤ 180, then 45 ≤ ½ θ ≤ 90, therefore, ½ θ is on quadrant I If 180 ≤ θ ≤ 270, then 90 ≤ ½ θ ≤ 135, therefore, ½ θ is on quadrant II If 270 ≤ θ ≤ 360, then 135 ≤ ½ θ ≤ 180, therefore, ½ θ is on quadrant II

1. Show that sin 90 = 1 by using sin 2(45 ) = 2 sin 45 cos 45 = 2. If cos A = -5/13 and A is the measure of an angle in quadrant II, find cos 2 A 3. If 180 < A < 270 and find

3. If cos x = 1/9 and x is in quad I, what is the value of sin Ans: 2/3 4. If sin A = –. 8, what is the value of cos 2 A? Ans: –. 28 5. If A = , what is the value of sin Ans: 2/5 6. If cos x = -3/5 and sin x < 0, find the value of sin 2 x Ans: 24/25