1. Homework Log Fri 4/22 Lesson 8 – 4 Learning Objective:
To use double & half angle identities to verify or find exact trig values
Hw: #810 Pg. 509 #4, 8, 10, 12, 16, 24, 28, 32, 33 – 39 odd, 42,
48, 54

2. 4/22/16 Lesson 8 – 4 Day 3 Double & Half Angle Identities
Advanced Math/Trig

3. Learning Objective
To use double or half angle identities to do the following:
Find exact Trig values
Rewrite Trig identities

4. 1. Find the exact values of all the trig functions of 2𝜃
tan 𝜃 =−2 2 & sec 𝜃 >0
Not Special Angle but can use pythag identity
1+ 𝑡𝑎𝑛 2 𝜃= 𝑠𝑒𝑐 2 𝜃
So cos 𝜃 = 1 3
1+ 𝑡𝑎𝑛 2 𝜃 = sec 𝜃
Since tan 𝜃 = sin 𝜃 cos 𝜃
1+ (−2 2 ) 2 = sec 𝜃
tan 𝜃 cos 𝜃 = sin 𝜃
− = sin 𝜃
1+4(2) = sec 𝜃
− = sin 𝜃
9 = sec 𝜃
3 = sec 𝜃

5. #1 Cont’d Find the exact values of all the trig functions of 2𝜃
sin 𝜃 =−
cos 𝜃 = 1 3
tan 𝜃 =−2 2
=2 −
= −
sin 2𝜃 =2 sin 𝜃 cos 𝜃
= − −
cos 2𝜃 = 𝑐𝑜𝑠 2 𝜃− 𝑠𝑖𝑛 2 𝜃
= 1 9 − 8 9
=− 7 9
sec 2𝜃 = 1 cos 2𝜃
=− 9 7 =−
csc 2𝜃 = 1 sin 2𝜃
= 9 −4 2

6. #1 Cont’d Find the exact values of all the trig functions of 2𝜃
sin 𝜃 =−
cos 𝜃 = 1 3
tan 𝜃 =−2 2
= − − 7 9
= −4 2 −7 =
tan 2𝜃 = sin 2𝜃 cos 2𝜃 =
cot 2𝜃 = 1 tan 2𝜃 =

7. 2. Find the exact values of sin, cos, and tan of 3𝜋 8
3𝜋 8 in QI
So cos 𝜃 >0 & sin 𝜃 >0
Use Half Angle Formulas!
3𝜋 8 = 𝜃 2 ?
3𝜋 8 =2𝜃 ?
3𝜋 16 =𝜃
6𝜋 8 =𝜃
cos 3𝜋 4 =−
NOT USEFUL
3𝜋 4 =𝜃
USEFUL

8. #2 Cont’d Find the exact values of sin, cos, and tan of 3𝜋 8
cos 3𝜋 4 =− cos 𝜃 >0 sin 𝜃 >0
sin 3𝜋 8 = sin 3𝜋 4 2
sin 𝜃 2 =± 1− cos 𝜃 2
= 1− − = = =

9. #2 Cont’d Find the exact values of sin, cos, and tan of 3𝜋 8
cos 3𝜋 4 =− cos 𝜃 >0 sin 𝜃 >0
= 1+ −
cos 3𝜋 8 = cos 3𝜋 4 2
cos 𝜃 2 =± 1+ cos 𝜃 2
= 2−
= 2− 2 4
= 2−

10. #2 Cont’d Find the exact values of sin, cos, and tan of 3𝜋 8
= −
tan 3𝜋 8 = sin 3𝜋 8 cos 3𝜋 8
= − 2
∙ 2− − 2
Multiply by denominator to get rid of big radical
= 4−2 2− 2
= − 2 ∙
= 2 (2+ 2 ) 4−2
Multiply by conjugate to get rid of radical =
= 2 +1

11. Ticket Out the Door
Determine the exact values of the sine, cosine, and tangent of 𝑜