Chapter 5: Analytic Trigonometry
Topic 2: Sum and Difference Formulas
Exact Values
Use any method you’d like to complete the charts below. Triangles and a unit circle are included to aid you.
𝟑𝟎°
𝟒𝟓°
𝟔𝟎°
𝟎°/𝟑𝟔𝟎°
𝟗𝟎°
𝟏𝟖𝟎°
𝐬𝐢𝐧 𝜽
𝐜𝐨𝐬 𝜽
𝐭𝐚𝐧 𝜽
𝟐𝟕𝟎°
𝐬𝐢𝐧 𝜽
𝐜𝐨𝐬 𝜽
𝐭𝐚𝐧 𝜽
Sum and Difference Formulas
These 6 reference sheet formulas can be used a variety of different ways. We will explore three of these uses.
Verifying known angle values
Using a sum or difference formula, verify the following values:
1. cos(90° − 60°) = cos 30°
2. sin(90° − 30°) = sin 60°
3. tan(90° − 45°) = tan 45°
4. sin(90° − 60°) = cos 60°
Recall: what other property is hidden in #4?
Calculating unknown angle values
Using a sum or difference formula, find the exact value.

Type one: Rewrite the angle as the sum or difference of two known angles.
5. cos 15°
6. sin 75°
7. sin
7𝜋
,
12
using the fact that
7𝜋
12
𝜋
3
= +
𝜋
4
5𝜋
5𝜋
𝜋

Type two: Find the equation that matches the pattern and work backwards.
9. cos 80° cos 20° + sin 80° sin 20°
10. cos 70° cos 40° + sin 70° sin 40°

Type three: Using given ratios to complete the question.
3
12
11. Given a quadrant II angle with sin 𝐴 = 5, and a quadrant IV angle with cos 𝐵 = 13, find:
a. cos(𝐴 + 𝐵)
b. sin(𝐴 − 𝐵)
𝜋
8. cos 12 , using the fact that 12 = 6 + 4
First: Finish the pieces
Verifying Identities
Use the skills from Topic #1 to verify each identity:
12.
cos(𝐴−𝐵)
sin 𝐴 cos 𝐵
= cot 𝐴 + tan 𝐵
Homework: Textbook Page 563
# 1 – 11, 17 – 31, 39 – 41… ALL ODD
13.
cos(𝐴−𝐵)
cos 𝐴 cos 𝐵
= 1 + tan 𝐴 tan 𝐵