Algebra 101 - Unit 16

Introduction

Double-angle and half-angle identities allow you to express trigonometric functions of multiples or fractions of an angle in terms of functions of the original angle. These formulas are useful for simplification, solving equations, and integration in calculus.

Double-Angle Formulas

• sin(2A) = 2 sin A cos A
• cos(2A) = cos² A − sin² A = 2 cos² A − 1 = 1 − 2 sin² A
• tan(2A) = (2 tan A) / (1 − tan² A)

Half-Angle Formulas

• sin²(A/2) = (1 − cos A)/2 → sin(A/2) = ±√((1 − cos A)/2)
• cos²(A/2) = (1 + cos A)/2 → cos(A/2) = ±√((1 + cos A)/2)
• tan(A/2) = sin A / (1 + cos A) = (1 − cos A)/sin A

Strategy for Using Formulas

1. Identify if the problem involves doubling or halving an angle.
2. Choose the appropriate identity for sine, cosine, or tangent.
3. Substitute known values and simplify carefully.
4. Check the sign (positive or negative) for half-angle formulas based on the quadrant.

Examples

Example 1 — Double Angle (Sine):

Find sin(2·30°): sin(2A) = 2 sin A cos A → 2·sin 30°·cos 30° = 2·(1/2)·(√3/2) = √3/2

Example 2 — Double Angle (Cosine):

Find cos(2·45°): cos(2A) = cos² A − sin² A → (√2/2)² − (√2/2)² = 0

Example 3 — Half Angle (Sine):

Find sin(15°): sin²(15°) = (1 − cos 30°)/2 → (1 − √3/2)/2 = (2 − √3)/4 → sin 15° = √((2 − √3)/4) ≈ 0.2588

Example 4 — Half Angle (Cosine):

Find cos(22.5°): cos²(22.5°) = (1 + cos 45°)/2 = (1 + √2/2)/2 → cos 22.5° = √((2 + √2)/4) ≈ 0.924

Practice Problems

1. Compute sin(60°) using the double-angle formula from sin(30°).
2. Compute cos(30°) using the half-angle formula from cos(60°).
3. Compute tan(75°) using tan(2A) formula with A = 37.5°.
4. Find sin(22.5°) using the half-angle formula from cos(45°).
5. Find cos(15°) using the half-angle formula from cos(30°).