Download Honors Analysis Section 7.7 Multiple Angle Identities

Honors Analysis
Section 7.7 Multiple Angle Identities
Q Problems
Evaluate each of the following. These values will be useful for later problems. (You may want to use the unit circle.)
Q1.) sin 30º
Q2.) cos 30º
Q3.) sin 45º
Q4.) cos 45º
Q5.) sin 60º
Q6.) cos 60º
Q7.) sin 90º
Q8.) cos 90º
Q9.) sin 135º
Q10.) cos 135º
Evaluate the exact value of each expression using sum/difference identities. Do not use a calculator (except to check
solutions), and show all work!
1.) cos 75°
2.) sin 15º
3.) sin 105º
4.) sin 165º
5.) cos 195 º
6.) cos(-15º)
7.) Explain how you could use trigonometric identities to determine tan 75º. Then evaluate.
8.) If 𝛼 and 𝛽 are first quadrant angles, and sin 𝛼 =
5
13
9.) If 𝛼 and 𝛽 are first quadrant angles, and tan 𝛼 =
4
3
4
and cos 𝛽 = 5, find cos (𝛼 + 𝛽).
and cot 𝛽 =
5
,
12
find sin (𝛼 − 𝛽).
Use the double angle identities below to answer the following questions:
sin 2𝜃 = 2 sin 𝜃 cos 𝜃
cos 2𝜃 = 2 𝑐𝑜𝑠 2 𝜃 − 1
cos 2𝜃 = 1 − 2 𝑠𝑖𝑛2 𝜃
10.) Use a 45º angle and a double angle identify to show that sin 90º = 1.
7
11. If cos 𝜃 = 25 and terminates in the first quadrant, determine:
A) sin 𝜃
B) sin 2𝜃
C) cos 2𝜃
D) tan 2𝜃
12. If tan y = 5/12 and y terminates in the third quadrant, find:
A) sin 2y
B) cos 2y
C) tan 2y
13.) Explain how you could use one of the above double angle identities to evaluate cos 22.5º (that is, HALF of a 45º
angle).
14.) Derive a formula for sin 3𝛼 in terms of 𝛼.