A-Level — mathsAdvanced2 min read

Compound and Double Angle Formulas

Apply compound angle and double angle identities at A-Level. Prove identities and solve equations.

Tutor AI Team
Tutor AI Team

Compound and double angle formulas express trig functions of sums and multiples of angles. They're essential for proving identities and solving trig equations at A-Level.

Compound Angle Formulas

sin⁡(A±B)=sin⁡Acos⁡B±cos⁡Asin⁡B\sin(A \pm B) = \sin A\cos B \pm \cos A\sin B cos⁡(A±B)=cos⁡Acos⁡B∓sin⁡Asin⁡B\cos(A \pm B) = \cos A\cos B \mp \sin A\sin B tan⁡(A±B)=tan⁡A±tan⁡B1∓tan⁡Atan⁡B\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A\tan B}

Double Angle Formulas

sin⁡2A=2sin⁡Acos⁡A\sin 2A = 2\sin A\cos A cos⁡2A=cos⁡2A−sin⁡2A=2cos⁡2A−1=1−2sin⁡2A\cos 2A = \cos^2 A - \sin^2 A = 2\cos^2 A - 1 = 1 - 2\sin^2 A tan⁡2A=2tan⁡A1−tan⁡2A\tan 2A = \frac{2\tan A}{1 - \tan^2 A}

Useful Rearrangements

cos⁡2A=1+cos⁡2A2\cos^2 A = \frac{1 + \cos 2A}{2} and sin⁡2A=1−cos⁡2A2\sin^2 A = \frac{1 - \cos 2A}{2} (useful for integration).

Worked Example: Example 1

Problem

Find exact value of cos⁡75°=cos⁡(45°+30°)\cos 75° = \cos(45° + 30°).

=cos⁡45°cos⁡30°−sin⁡45°sin⁡30°=22⋅32−22⋅12=6−24= \cos 45°\cos 30° - \sin 45°\sin 30° = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2} = \frac{\sqrt{6} - \sqrt{2}}{4}

Solution

Worked Example: Example 2

Problem

Solve cos⁡2x=1−3sin⁡x\cos 2x = 1 - 3\sin x for 0≤x≤2π0 \leq x \leq 2\pi.

1−2sin⁡2x=1−3sin⁡x1 - 2\sin^2 x = 1 - 3\sin x → 2sin⁡2x−3sin⁡x=02\sin^2 x - 3\sin x = 0 → sin⁡x(2sin⁡x−3)=0\sin x(2\sin x - 3) = 0.

sin⁡x=0\sin x = 0 → x=0,π,2πx = 0, \pi, 2\pi. sin⁡x=32\sin x = \frac{3}{2} — impossible.

Solution

Practice Problems

    1. Prove sin⁡2A1+cos⁡2A=tan⁡A\frac{\sin 2A}{1 + \cos 2A} = \tan A.
    1. Find the exact value of sin⁡15°\sin 15°.
    1. Solve 3cos⁡2θ+sin⁡θ=13\cos 2\theta + \sin\theta = 1 for 0≤θ≤360°0 \leq \theta \leq 360°.

Want to check your answers and get step-by-step solutions?

Get it on Google PlayDownload on the App Store

Key Takeaways

  • ✓

    Compound: sin⁡(A+B),cos⁡(A+B),tan⁡(A+B)\sin(A+B), \cos(A+B), \tan(A+B).

  • ✓

    Double angle: set B=AB = A in compound formulas.

  • ✓

    cos⁡2A\cos 2A has three forms — choose based on context.

  • ✓

    These are used for proving identities, solving equations, and integration.

Tags:
#a-level#maths#pure#trigonometry#identities

Related Topics

Ready to Ace Your A-Level maths?

Get instant step-by-step solutions to any problem. Snap a photo and learn with Tutor AI — your personal exam prep companion.

Get it on Google PlayDownload on the App Store