1. 5.5 Multiple Angle & Product-to-Sum Formulas
Homework: Page 394, #3, 5, 7, 14, 32, 33, 40, 48, 57

2. Double Angle Formulas

3. Double Angle Formulas

4. Double Angle Formulas

5. Power Reducing Formulas

6. Half-Angle Formulas
The signs of sine and cosine depend on which quadrant the angle ends up in.

7. Product-to-Sum Formulas

8. Sum-to-Product Formulas

9. Examples Example 1: Solve 𝑐𝑜𝑠2𝑥+𝑐𝑜𝑠𝑥=0 2 𝑐𝑜𝑠 2 𝑥−1+𝑐𝑜𝑠𝑥=0
2 𝑐𝑜𝑠 2 𝑥+𝑐𝑜𝑠𝑥−1=0
2𝑐𝑜𝑠𝑥−1 𝑐𝑜𝑠𝑥+1 =0
2𝑐𝑜𝑠𝑥−1 =0 𝑎𝑛𝑑 𝑐𝑜𝑠𝑥+1 =0
𝑐𝑜𝑠𝑥= 1 2 𝑎𝑛𝑑 𝑐𝑜𝑠𝑥=−1
𝑥= 𝑐𝑜𝑠 − 𝑎𝑛𝑑 𝑥= 𝑐𝑜𝑠 −1 −1
𝑥= 𝜋 3 +2𝑛𝜋, 𝑥= 5𝜋 3 +2𝑛𝜋
𝑥=𝜋+2𝑛𝜋

10. Example 2: Analyze the graph on the interval [0, 2π)
𝑦=3(1− 2𝑠𝑖𝑛 2 𝑥)
𝑦=3𝑐𝑜𝑠(2𝑢)
Amplitude = 3, period = π
Key points on the interval [0,π] are:
0, 3 , 𝜋 4 , 0 , 𝜋 2 , −3 , 3𝜋 4 , 0 , (𝜋,3)

11. Example 3: Find sin(2u), cos(2u), and tan(2u), given:
𝑠𝑖𝑛𝑢= 3 5 , 0<𝑢< 𝜋 2
𝑐𝑜𝑠𝑢= 4 5
𝑡𝑎𝑛𝑢= 3 4
tan 2𝑢 = 2∙ −
cos 2𝑢 = −
sin 2𝑢 =2∙ ∙ 4 5
= 24 25
= − 9 15
= − 9 16
= 7 25
= 3 2 ∙ 16 7
= 24 7

12. Example 4: Derive a triple-angle formula for cos3x
cos 3𝑥 =cos⁡(2𝑥+𝑥)
=𝑐𝑜𝑠2𝑥𝑐𝑜𝑠𝑥−𝑠𝑖𝑛2𝑥𝑠𝑖𝑛𝑥
= 2 𝑐𝑜𝑠 2 𝑥−1 𝑐𝑜𝑠𝑥−(2𝑠𝑖𝑛𝑥𝑐𝑜𝑠𝑥)𝑠𝑖𝑛𝑥
=2 𝑐𝑜𝑠 3 𝑥−𝑐𝑜𝑠𝑥−2 𝑠𝑖𝑛 2 𝑥𝑐𝑜𝑠𝑥
=2 𝑐𝑜𝑠 3 𝑥−𝑐𝑜𝑠𝑥−2𝑐𝑜𝑠𝑥 (1−𝑐𝑜𝑠 2 𝑥)
=2 𝑐𝑜𝑠 3 𝑥−𝑐𝑜𝑠𝑥−2𝑐𝑜𝑠𝑥+ 2 𝑐𝑜𝑠 3 𝑥
=4 𝑐𝑜𝑠 3 𝑥−3𝑐𝑜𝑠𝑥

13. Example 5: Rewrite tan4x as a quotient of first powers of the cosines of multiple angles.
𝑡𝑎𝑛 4 𝑥= 1−𝑐𝑜𝑠2𝑥 1+cos⁡2𝑥 1−𝑐𝑜𝑠2𝑥 1+cos⁡2𝑥
= 1−𝑐𝑜𝑠2𝑥−𝑐𝑜𝑠2𝑥+ 𝑐𝑜𝑠 2 2𝑥 1+𝑐𝑜𝑠2𝑥+𝑐𝑜𝑠2𝑥+ 𝑐𝑜𝑠 2 2𝑥
= 1−2𝑐𝑜𝑠2𝑥+ 1+𝑐𝑜𝑠4𝑥 𝑐𝑜𝑠2𝑥+ 1+𝑐𝑜𝑠4𝑥 2
= 2−4𝑐𝑜𝑠2𝑥+1+𝑐𝑜𝑠4𝑥 𝑐𝑜𝑠2𝑥+1+𝑐𝑜𝑠4𝑥 2
= 3−4𝑐𝑜𝑠2𝑥+𝑐𝑜𝑠4x 3+4𝑐𝑜𝑠2𝑥+𝑐𝑜𝑠4𝑥

14. Example 6: Use the half-angle formulas to determine the exact values of the sine, cosine, and tangent of 15.
NOTE: 15 is half of 30 and 15 lies in Quadrant I:
𝑠𝑖𝑛 30° 2 = 1−𝑐𝑜𝑠30 2
𝑐𝑜𝑠 30° 2 = 1+𝑐𝑜𝑠30 2
= 1− =
= 2− =
= 2− ∙ 1 2
= ∙ 1 2
= 2− =

15. 𝑡𝑎𝑛 30° 2 = 1−𝑐𝑜𝑠30° 𝑠𝑖𝑛30°
= 1−
= 2−
= 2−  2 1
= 2− 3 2

16. Example 7: Rewrite the following product as a sum or difference:
𝑠𝑖𝑛5𝑥𝑐𝑜𝑠3𝑥
= sin 5𝑥+3𝑥 +sin⁡(5𝑥−3𝑥)
= sin 8𝑥 +sin⁡(2𝑥)