1. 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, p.511 # 10,12,14

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

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

4. Double angle identities
sin 2A = 2 sin A cos A
cos 2A = cos2 A – sin2 A
cos 2A = 2cos2 A – 1
cos 2A = 1 – 2sin2 A

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

6. If 0 ≤ θ ≤ 90, then 0 ≤ ½ θ ≤ 45, 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

7. 1. Show that sin 90 = 1 by using sin 2(45)
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 2A
3. If 180 < A < 270 and find

8.  x is in quad I, what is the value of sin
3. If cos x = 1/9 and
3. If cos x = 1/9 and
3. If cos x = 1/9 and
 x is in quad I, what is the value of sin
 x is in quad I, what is the value of sin
 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 2A?
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 2x
Ans: 24/25