Double Angle Formulae: Concepts, Formulas and Examples

The double-angle formulae give the sine, cosine and tangent of 2θ in terms of θ: sin 2θ = 2 sin θ cos θ, cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ, and tan 2θ = 2tan θ/(1 − tan²θ). They come from setting B = A in the sum formulae.

They matter because doubling appears everywhere: the range of a projectile is R = v²sin 2θ/g, alternating current oscillates at twice the phase rate, and calculus uses cos 2θ to integrate powers of sine and cosine.

Key Concepts

• Derivation: put B = θ in sin(θ + θ), cos(θ + θ) and tan(θ + θ) to get each double-angle formula instantly.
• Three forms of cos 2θ: cos²θ − sin²θ, 2cos²θ − 1, and 1 − 2sin²θ are all equal; pick the one matching the data you have.
• Power reduction: the forms 2cos²θ − 1 and 1 − 2sin²θ convert squared functions into first powers, which is exactly what integration needs.
• Not doubling the value: sin 2θ is not 2 sin θ; the cosine factor keeps it within [−1, 1].

Formulas

• Sine: sin 2θ = 2 sin θ cos θ.
• Cosine: cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ.
• Tangent: tan 2θ = 2tan θ/(1 − tan²θ), valid where tan θ is defined and tan²θ ≠ 1.
• Assumptions: θ is any real angle; choose the cosine form that uses the ratio you already know.

Worked Example

If sin θ = 3/5 and θ is acute, find sin 2θ and cos 2θ.

1. Find cos θ: cos²θ = 1 − 9/25 = 16/25, and θ acute gives cos θ = 4/5.
2. sin 2θ = 2 sin θ cos θ = 2 × (3/5) × (4/5) = 24/25.
3. cos 2θ = cos²θ − sin²θ = 16/25 − 9/25 = 7/25.
4. Verify with the identity: (24/25)² + (7/25)² = (576 + 49)/625 = 625/625 = 1, as sin²2θ + cos²2θ must equal.
5. Cross-check with another form: 1 − 2sin²θ = 1 − 2(9/25) = 1 − 18/25 = 7/25. Both cosine forms agree.

Common Misconceptions

• sin 2θ ≠ 2 sin θ. With θ = 30°, sin 60° = √3/2 ≈ 0.866 while 2 sin 30° = 1. The 2 sin θ cos θ form is the correct one.
• All three cosine forms are the same identity. They are not three separate rules to memorize independently; they follow from sin²θ + cos²θ = 1.
• tan 2θ has its own domain. It fails when tan²θ = 1, i.e. θ = 45° + n·90°, because the denominator vanishes.

Real-World Applications

• Projectile motion: maximum range formulas use sin 2θ, with the best launch angle at 45°.
• Calculus: integrating sin²θ and cos²θ via the power-reduction forms.
• Electrical engineering: power in AC circuits involves cos 2ωt terms from squaring sinusoidal voltage.
• Oscillations: energy expressions in simple harmonic motion simplify with double-angle forms.

The double-angle formulae — sin 2θ = 2 sin θ cos θ, three equivalent forms of cos 2θ, and tan 2θ = 2tan θ/(1 − tan²θ) — all come from the sum formulae with B = θ. Choose the cosine form that fits your known data, and verify with sin²2θ + cos²2θ = 1.