1. Homework Log Mon 4/18 Lesson 8 – 3 Learning Objective:
To use sine & tangent of sum or difference angles to verify or find exact trig values
Hw: #806 Pg – 31 odd

2. 4/18/16 Lesson 8 – 3 Day 1 Sin & Tan of the Diff or Sum of Two Angles
Advanced Math/Trig

3. Learning Objective
To use sine & tangent of the sum or difference of angles to
Find exact Trig values
Rewrite Trig identities

4. Sum & Difference Equations
sin (𝛼+𝛽) = sin 𝛼 cos 𝛽 + cos 𝛼 sin 𝛽
sin (𝛼−𝛽) = sin 𝛼 cos 𝛽 − cos 𝛼 sin 𝛽
Memorize!!
tan (𝛼+𝛽) = tan 𝛼 + tan 𝛽 1− tan 𝛼 tan 𝛽
tan (𝛼−𝛽) = tan 𝛼 − tan 𝛽 1+ tan 𝛼 tan 𝛽

5. Find the Exact Value 1. sin 20 𝑜 cos 110 𝑜 − cos 20 𝑜 sin 110 𝑜
=−1
2. sin 115 𝑜 cos 20 𝑜 + cos 115 𝑜 sin 20 𝑜
= sin ( 115 𝑜 + 20 𝑜 ) =
= sin ( 135 𝑜 )

6. Find the Exact Value 3. tan 43 𝑜 + tan 137 𝑜 1− tan 43 𝑜 tan 137 𝑜=0
= tan ( −240 𝑜 )
=− tan ( 240 𝑜 )
=− 3

7. Find the exact value of tan (𝛼−𝛽)
5. Given: tan α=1.1 & tan 𝛽=3.7
tan (𝛼−𝛽) = tan 𝛼 − tan 𝛽 1+ tan 𝛼 tan 𝛽
tan (𝛼−𝛽) = 1.1−3.7 1+(1.1)(3.7)
= −
=− 20 39

8. Find the exact value of tan (𝛼+𝛽)
6. Given: cot α=0.7 & cot 𝛽=1.4
tan α= 1 0.7
= 10 7
tan 𝛽= 1 1.4
= 5 7
tan (𝛼+𝛽) = tan 𝛼 + tan 𝛽 1− tan 𝛼 tan 𝛽
tan (𝛼+𝛽) = −
= − 50 49
= − 1 49
=−105

9. 7. Find the exact value of sin 165 𝑜
sin 165 𝑜 = sin ( 120 𝑜 + 45 𝑜 )
= sin 120 𝑜 cos 45 𝑜 + cos 120 𝑜 sin 45 𝑜
= −
= 6 − 2 4

10. 8. Find the exact value of sin − 7𝜋 12
sin − 7𝜋 12 = −sin 7𝜋 12
=− sin 𝜋 3 + 𝜋 4
=− sin 𝜋 3 cos 𝜋 cos 𝜋 3 sin 𝜋 4 =−
= − 6 − 2 4

11. 9. Find the exact value of tan 5𝜋 12
= tan 𝜋 6 + tan 𝜋 4 1− tan 𝜋 6 tan 𝜋 4
tan 5𝜋 12
= tan 𝜋 6 + 𝜋 4
= −
= −
= − 3 3
= ∙ 3 3− 3
= − 3 ∙
=2+ 3

12. Express the trig function in terms of 𝜃
10. sin (𝜃+ 3𝜋 4 )
= sin 𝜃 cos 3𝜋 4 + cos 𝜃 sin 3𝜋 4
= sin 𝜃 − cos 𝜃
=− sin 𝜃 cos 𝜃

13. Express the trig function in terms of 𝜃
= tan 𝜃 − tan 3𝜋 1+ tan 𝜃 tan 3𝜋
11. tan (𝜃−3𝜋 )
= tan 𝜃 −0 1+ tan 𝜃 (0)
= tan 𝜃

14. Find sin (𝛼−𝛽) and tan (𝛼+𝛽)
12. Given: sin α= & sin 𝛽= 5 13
𝛼 in QII and 𝛽 in QI
Need to find sin 𝛼 , tan 𝛼 & cos 𝛽 , tan 𝛽
For 𝛼, y = 15 & r = 17
For 𝛽, y = 5 & r = 13
𝑥 2 + (15) 2 = 17 2
𝑥 2 + (5) 2 = (13) 2
𝑥=−8
QII
𝑥=12 QI
cos𝛼=− 8 17
tan𝛼=− 15 8
cos𝛽= 12 13
tan𝛽= 5 12

15. #12 Cont’d Find sin (𝛼−𝛽) sin (𝛼−𝛽) = sin 𝛼 cos 𝛽 − cos 𝛼 sin 𝛽
cos α=− sin𝛼= tan 𝛼 =− 15 8
cos 𝛽= sin𝛽= tan 𝛽 = 5 12
sin (𝛼−𝛽) = sin 𝛼 cos 𝛽 − cos 𝛼 sin 𝛽
= − − = =

16. #12 Cont’d Find tan (𝛼+𝛽) = − 15 8 + 5 12 1− − 15 8 5 12
cos α=− sin𝛼= tan 𝛼 =− 15 8
cos 𝛽= sin𝛽= tan 𝛽 = 5 12
= −
= − − −
= tan 𝛼 + tan 𝛽 1− tan 𝛼 tan 𝛽 =−

17. Ticket Out the Door
Find exact value of tan 285 𝑜