Multiple Angle Formulas

[MathDefinition title="Double Angle Formulas"] Double angle formulas express trigonometric functions of 2A in terms of functions of angle A. These are derived from compound angle formulas by setting B = A. [/MathDefinition]

[MathKeyFormula title="Double Angle Formulas"]

1. sin 2A = 2 sin A cos A
2. cos 2A = cos² A − sin² A = 2cos² A − 1 = 1 − 2sin² A
3. tan 2A = 2 tan A / (1 − tan² A)
4. sin 2A = 2 tan A / (1 + tan² A)
5. cos 2A = (1 − tan² A) / (1 + tan² A) [/MathKeyFormula]

[MathNote title="When to Use Each Form"]

• Use sin 2A = 2 sin A cos A when you have both sin A and cos A
• Use cos 2A = 2cos² A − 1 when you know cos A
• Use cos 2A = 1 − 2sin² A when you know sin A [/MathNote]

[MathExample title="Example 1: Finding sin 2A"] Problem: If sin A = 3/5 and A is in the first quadrant, find sin 2A.

Solution: Given sin A = 3/5, we need to find cos A. cos² A = 1 − sin² A = 1 − (3/5)² = 1 − 9/25 = 16/25 cos A = 4/5 (since A is in first quadrant)

sin 2A = 2 sin A cos A = 2 × (3/5) × (4/5) = 24/25

Therefore, sin 2A = 24/25 [/MathExample]

[MathExample title="Example 2: Finding cos 2A"] Problem: If cos A = 5/13, find cos 2A.

Solution: Given cos A = 5/13

Using cos 2A = 2cos² A − 1: cos 2A = 2(5/13)² − 1 = 2(25/169) − 1 = 50/169 − 169/169 = −119/169

Therefore, cos 2A = −119/169 [/MathExample]

[MathExample title="Example 3: Proving Identity"] Problem: Prove that cos 2A = (1 − tan² A) / (1 + tan² A).

Solution: RHS = (1 − tan² A) / (1 + tan² A) = (1 − sin² A/cos² A) / (1 + sin² A/cos² A) = (cos² A − sin² A)/cos² A / (cos² A + sin² A)/cos² A = (cos² A − sin² A) / (cos² A + sin² A) = (cos² A − sin² A) / 1 = cos² A − sin² A = cos 2A = LHS

Hence proved. [/MathExample]

[MathExample title="Example 4: Expressing in Single Term"] Problem: Express 2 sin 30° cos 30° as a single trigonometric ratio.

Solution: We know that 2 sin A cos A = sin 2A

Here A = 30° 2 sin 30° cos 30° = sin(2 × 30°) = sin 60°

Verification: 2 sin 30° cos 30° = 2 × (1/2) × (√3/2) = √3/2 sin 60° = √3/2

Therefore, 2 sin 30° cos 30° = sin 60° = √3/2 [/MathExample]