1. Warm Up
Express each angle (a) as a sum and (b) as a difference of multiples of 30°, 45°, 𝑜𝑟 60° ° ° ° Express each angle (a) as a sum and (b) as a difference of multiples of 𝜋 6 , 𝜋 4 , 𝑜𝑟 𝜋 𝜋 − 𝜋 12

2. Chapter 10: Trigonometric Addition Formulas (Identities)
Simplify Expressions
Evaluate Expressions
Prove Equations
Solve Equations

3. F F F T F True or False? 𝑥+4 2 = 𝑥 2 +16 − 4 2 =16
𝑥+4 2 = 𝑥 2 +16
− 4 2 =16
𝑥 2 𝑥 2 −3𝑦 = 1 1−3𝑦
100% =100%
cos 10°+30° =𝑐𝑜𝑠10°+𝑐𝑜𝑠30° F F T F

4. cos 10°+30° =𝑐𝑜𝑠10°+𝑐𝑜𝑠30°
cos 40° =𝑐𝑜𝑠10°+𝑐𝑜𝑠30°
0.77=
0.77=1.85 ≠

5. Section 10.1 Formulas for 𝑐𝑜𝑠 𝛼±𝛽 and for 𝑠𝑖𝑛 𝛼±𝛽
Chapter 10 Trigonometric Addition Formulas (Identities)
Section Formulas for 𝑐𝑜𝑠 𝛼±𝛽 and for 𝑠𝑖𝑛 𝛼±𝛽

6. by Michael Stueben from Twenty Years Before the Blackboard
Sinbad and Cosette
A Love Story?
by Michael Stueben from Twenty Years Before the Blackboard

7. Sinbad loved Cosette ….

8. …but Cosette did not feel the same way about Sinbad.

9. Sinbad and Cosette’s parents were friends and the two had to go out on dates.

10. In order for it not to be too painful, they brought their siblings.

11. Naturally, when Sinbad was in charge of their double date, he put himself with Cosette, and he put his brother with her sister:

12. Sinbad loved to tell people that his and Cosette's signs were the same.

13. sin(𝛼 + 𝛽) = sin 𝛼 cos 𝛽 + cos 𝛼 sin 𝛽

14. However, when Cosette was in charge of the double date she placed herself with her sister and put Sinbad with his brother.

15. She made sure everyone knew that their signs were NOT the same.

16. cos(𝛼 + 𝛽) = cos 𝛼 cos 𝛽 - sin 𝛼 sin 𝛽
Also, notice that Cosette placed herself and her sister BEFORE Sinbad and his brother. This detail was important to Cosette. She was very snobby, you know.

17. And that is the story of Sinbad and Cosette……
THE END

18. Sum & Difference Formulas

19. Classwork Problems #1-11
Work on the problems one at a time in your group.
Check your answer with everyone else’s. Make sure you all have the SAME answer.
Move on to the next problem.

20. Classwork A. Simplify. (Write as a single trig ratio first)
1. 𝑠𝑖𝑛 𝜋 6 𝑐𝑜𝑠 𝜋 2 +𝑐𝑜𝑠 𝜋 6 𝑠𝑖𝑛 𝜋 𝑐𝑜𝑠 𝜋 3 𝑐𝑜𝑠 𝜋 6 +𝑠𝑖𝑛 𝜋 3 𝑠𝑖𝑛 𝜋 6
3. 𝑐𝑜𝑠𝑥𝑐𝑜𝑠1−𝑠𝑖𝑛𝑥𝑠𝑖𝑛1
B. Evaluate.
4. 𝑠𝑖𝑛 𝜋 3 + 𝜋 cos⁡(45°−60°) 6. 𝑠𝑖𝑛 𝜋 2 − 𝜋 6
7. 𝑠𝑖𝑛 𝜋+𝑥 + 𝑠𝑖𝑛 𝜋−𝑥
C. Find the exact value 𝑐𝑜𝑠15° 9. 𝑠𝑖𝑛 5𝜋 𝑠𝑖𝑛195°

21. Simplify. (Write as a single trig ratio first) 1
=𝑠𝑖𝑛 𝜋 6 + 𝜋 2
=𝑠𝑖𝑛 4𝜋 6
= 𝟑 𝟐
=𝑠𝑖𝑛 𝜋 6 + 3𝜋 6
=𝒔𝒊𝒏 𝟐𝝅 𝟑
=𝑐𝑜𝑠 𝜋 3 − 𝜋 6
2. 𝑐𝑜𝑠 𝜋 3 𝑐𝑜𝑠 𝜋 6 +𝑠𝑖𝑛 𝜋 3 𝑠𝑖𝑛 𝜋 6
= 𝟑 𝟐
=𝒄𝒐𝒔 𝝅 𝟔
3. 𝑐𝑜𝑠𝑥𝑐𝑜𝑠1−𝑠𝑖𝑛𝑥𝑠𝑖𝑛1
=𝑐𝑜𝑠 𝑥+1

22. Evaluate. 4. 𝑠𝑖𝑛 𝜋 3 + 𝜋 4 5. cos⁡(45°−60°) =𝑐𝑜𝑠45°𝑐𝑜𝑠60°+𝑠𝑖𝑛45°𝑠𝑖𝑛60°
=𝑠𝑖𝑛 𝜋 3 𝑐𝑜𝑠 𝜋 4 +𝑐𝑜𝑠 𝜋 3 𝑠𝑖𝑛 𝜋 4
= ∙ ∙ =
= 𝟔 + 𝟐 𝟒
5. cos⁡(45°−60°)
=𝑐𝑜𝑠45°𝑐𝑜𝑠60°+𝑠𝑖𝑛45°𝑠𝑖𝑛60°
= ∙ ∙ =
= 𝟐 + 𝟔 𝟒

23. Evaluate. 6. 𝑠𝑖𝑛 𝜋 2 − 𝜋 6 7. 𝑠𝑖𝑛 𝜋+𝑥 + 𝑠𝑖𝑛 𝜋−𝑥
=𝑠𝑖𝑛 𝜋 2 𝑐𝑜𝑠 𝜋 6 −𝑐𝑜𝑠 𝜋 2 𝑠𝑖𝑛 𝜋 6
=1∙ ∙ 1 2
= 𝟑 𝟐
7. 𝑠𝑖𝑛 𝜋+𝑥 + 𝑠𝑖𝑛 𝜋−𝑥
=𝑠𝑖𝑛𝜋𝑐𝑜𝑠𝑥+𝑐𝑜𝑠𝜋𝑠𝑖𝑛𝑥+𝑠𝑖𝑛𝜋𝑐𝑜𝑠𝑥−𝑐𝑜𝑠𝜋𝑠𝑖𝑛𝑥
=𝑠𝑖𝑛𝜋𝑐𝑜𝑠𝑥+𝑠𝑖𝑛𝜋𝑐𝑜𝑠𝑥
=2𝑠𝑖𝑛𝜋𝑐𝑜𝑠𝑥
=2∙0∙𝑐𝑜𝑠𝑥 =𝟎

24. Find the exact value. 8. 𝑐𝑜𝑠15°
=𝑐𝑜𝑠 45°−30°
=𝑐𝑜𝑠45°𝑐𝑜𝑠30°+𝑠𝑖𝑛45°𝑠𝑖𝑛30°
= ∙ ∙ 1 2
= 𝟔 + 𝟐 𝟒
=𝑠𝑖𝑛 3𝜋 4 − 𝜋 3
9. 𝑠𝑖𝑛 5𝜋 12
=𝑠𝑖𝑛 3𝜋 4 𝑐𝑜𝑠 𝜋 3 −𝑐𝑜𝑠 3𝜋 4 𝑠𝑖𝑛 𝜋 3
= ∙ 1 2 − − ∙
= 𝟐 + 𝟔 𝟒

25. Find the exact value. 10. 𝑠𝑖𝑛195°
=𝑠𝑖𝑛 135°+60°
=𝑠𝑖𝑛135°𝑐𝑜𝑠60°+𝑐𝑜𝑠135°𝑠𝑖𝑛60°
= ∙ − ∙
= 𝟐 − 𝟔 𝟒

26. 11. Evaluate. Suppose that 𝑠𝑖𝑛𝛼= and 𝑠𝑖𝑛𝛽= where 0<𝛼< 𝜋 2 and 𝜋 2 <𝛽<𝜋 Find 𝒄𝒐𝒔 𝜶+𝜷
cos(𝛼 + 𝛽) = cos 𝛼 cos 𝛽 - sin 𝛼 sin 𝛽

28. Homework
Page 373
#1-5,13-31 odds

29. 12. 𝑐𝑜𝑠 𝑥+𝑦 −𝑐𝑜𝑠 𝑥−𝑦 Simplify. 𝑐𝑜𝑠𝑥𝑐𝑜𝑠𝑦−𝑠𝑖𝑛𝑥𝑠𝑖𝑛𝑦− 𝑐𝑜𝑠𝑥𝑐𝑜𝑠𝑦+𝑠𝑖𝑛𝑥𝑠𝑖𝑛𝑦
12. 𝑐𝑜𝑠 𝑥+𝑦 −𝑐𝑜𝑠 𝑥−𝑦
𝑐𝑜𝑠𝑥𝑐𝑜𝑠𝑦−𝑠𝑖𝑛𝑥𝑠𝑖𝑛𝑦− 𝑐𝑜𝑠𝑥𝑐𝑜𝑠𝑦+𝑠𝑖𝑛𝑥𝑠𝑖𝑛𝑦
𝑐𝑜𝑠𝑥𝑐𝑜𝑠𝑦−𝑠𝑖𝑛𝑥𝑠𝑖𝑛𝑦−𝑐𝑜𝑠𝑥𝑐𝑜𝑠𝑦−𝑠𝑖𝑛𝑥𝑠𝑖𝑛𝑦
−𝑠𝑖𝑛𝑥𝑠𝑖𝑛𝑦−𝑠𝑖𝑛𝑥𝑠𝑖𝑛𝑦
−2𝑠𝑖𝑛𝑥𝑠𝑖𝑛𝑦

30. 𝑠𝑖𝑛𝑥𝑠𝑖𝑛𝑦𝑐𝑜𝑡𝑥𝑐𝑜𝑡𝑦−𝑠𝑖𝑛𝑥𝑠𝑖𝑛𝑦
Simplify.
𝑠𝑖𝑛𝑥𝑠𝑖𝑛𝑦 𝑐𝑜𝑡𝑥𝑐𝑜𝑡𝑦−1
𝑠𝑖𝑛𝑥𝑠𝑖𝑛𝑦𝑐𝑜𝑡𝑥𝑐𝑜𝑡𝑦−𝑠𝑖𝑛𝑥𝑠𝑖𝑛𝑦
𝑠𝑖𝑛𝑥𝑠𝑖𝑛𝑦 𝑐𝑜𝑠𝑥 𝑠𝑖𝑛𝑥 𝑐𝑜𝑠𝑦 𝑠𝑖𝑛𝑦 −𝑠𝑖𝑛𝑥𝑠𝑖𝑛𝑦
𝑐𝑜𝑠𝑥𝑐𝑜𝑠𝑦−𝑠𝑖𝑛𝑥𝑠𝑖𝑛𝑦
𝑐𝑜𝑠 𝑥+𝑦

31. 14. Prove: sin 3𝜋 2 −𝑥 =−𝑐𝑜𝑠𝑥 𝑠𝑖𝑛 3𝜋 2 𝑐𝑜𝑠𝑥−𝑐𝑜𝑠 3𝜋 2 sinx
Sine Sum Identity
−1 𝑐𝑜𝑠𝑥− 0 sinx
Evaluate
−𝑐𝑜𝑠𝑥
Simplify
−𝑐𝑜𝑠𝑥=−𝑐𝑜𝑠𝑥

32. 15. Prove: 𝑐𝑜𝑠 𝑥+𝑦 +𝑐𝑜𝑠 𝑦−𝑥 =2𝑐𝑜𝑠𝑥𝑐𝑜𝑠𝑦
𝑐𝑜𝑠𝑥𝑐𝑜𝑠𝑦−𝑠𝑖𝑛𝑥𝑠𝑖𝑛𝑦+ 𝑐𝑜𝑠𝑦𝑐𝑜𝑠𝑥+𝑠𝑖𝑛𝑦𝑠𝑖𝑛𝑥
Cosine Sum & Difference Identities
𝑐𝑜𝑠𝑥𝑐𝑜𝑠𝑦+𝑐𝑜𝑠𝑦𝑐𝑜𝑠𝑥
Add
2𝑐𝑜𝑠𝑥𝑐𝑜𝑠𝑦=2𝑐𝑜𝑠𝑥𝑐𝑜𝑠𝑦
Add

33. 16. 𝑠𝑖𝑛 𝜋 3 +𝑥 −𝑠𝑖𝑛 𝜋 3 −𝑥 =1 Solve for 𝟎≤𝒙<𝟐𝝅
16. 𝑠𝑖𝑛 𝜋 3 +𝑥 −𝑠𝑖𝑛 𝜋 3 −𝑥 =1
𝑠𝑖𝑛 𝜋 3 𝑐𝑜𝑠𝑥+𝑐𝑜𝑠 𝜋 3 𝑠𝑖𝑛𝑥− 𝑠𝑖𝑛 𝜋 3 𝑐𝑜𝑠𝑥−𝑐𝑜𝑠 𝜋 3 𝑠𝑖𝑛𝑥 =1
3 2 𝑐𝑜𝑠𝑥+ 1 2 𝑠𝑖𝑛𝑥− 𝑐𝑜𝑠𝑥+ 1 2 𝑠𝑖𝑛𝑥=1
1 2 𝑠𝑖𝑛𝑥+ 1 2 𝑠𝑖𝑛𝑥=1
𝑥= 𝜋 2
𝑠𝑖𝑛𝑥=1