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Multiple-Angle Formulas — Definition, Formula & Examples

Multiple-angle formulas are trigonometric identities that rewrite functions of nθn\theta (like sin⁡3θ\sin 3\theta or cos⁡4θ\cos 4\theta) using only sin⁡θ\sin\theta and cos⁡θ\cos\theta. The double-angle and triple-angle formulas are the most commonly used cases.

A multiple-angle formula expresses sin⁡(nθ)\sin(n\theta), cos⁡(nθ)\cos(n\theta), or tan⁡(nθ)\tan(n\theta) as a polynomial in sin⁡θ\sin\theta and cos⁡θ\cos\theta (or rational expression for tangent), derived by repeated application of the sum/difference identities. These are sometimes called Chebyshev-type expansions for cosine.

Key Formula

sin⁡3θ=3sin⁡θ−4sin⁡3θ\sin 3\theta = 3\sin\theta - 4\sin^3\theta cos⁡3θ=4cos⁡3θ−3cos⁡θ\cos 3\theta = 4\cos^3\theta - 3\cos\theta tan⁡3θ=3tan⁡θ−tan⁡3θ1−3tan⁡2θ\tan 3\theta = \frac{3\tan\theta - \tan^3\theta}{1 - 3\tan^2\theta}
Where:
  • θ\theta = Any angle measure (radians or degrees)

How It Works

Start with the sum identity: sin⁡(A+B)=sin⁡Acos⁡B+cos⁡Asin⁡B\sin(A+B) = \sin A\cos B + \cos A\sin B. To get a triple-angle formula, set A=2θA = 2\theta and B=θB = \theta, then substitute the double-angle results for sin⁡2θ\sin 2\theta and cos⁡2θ\cos 2\theta. Each higher multiple builds on the previous ones. In practice, you only need to memorize the double- and triple-angle cases, since higher multiples can always be derived the same way.

Worked Example

Problem: Find the exact value of cos⁡3θ\cos 3\theta when θ=20°\theta = 20°.
Step 1: Identify the target angle: 3×20°=60°3 \times 20° = 60°.
cos⁡3(20°)=cos⁡60°\cos 3(20°) = \cos 60°
Step 2: Apply the triple-angle formula and set it equal to the known value.
4cos⁡320°−3cos⁡20°=cos⁡60°=124\cos^3 20° - 3\cos 20° = \cos 60° = \frac{1}{2}
Step 3: This equation can be used to solve for cos⁡20°\cos 20°, which has no simple radical form. The formula confirmed that cos⁡60°\cos 60° can be expressed through cos⁡20°\cos 20°.
4cos⁡320°−3cos⁡20°=124\cos^3 20° - 3\cos 20° = \frac{1}{2}
Answer: The triple-angle formula gives 4cos⁡320°−3cos⁡20°=124\cos^3 20° - 3\cos 20° = \dfrac{1}{2}, confirming the identity and relating cos⁡20°\cos 20° to the known value cos⁡60°=12\cos 60° = \frac{1}{2}.

Why It Matters

Multiple-angle formulas appear in solving trigonometric equations where the argument is nθn\theta, in Fourier analysis, and in signal processing. In calculus, they help reduce powers of trig functions for integration (e.g., rewriting sin⁡3x\sin^3 x using the triple-angle identity).

Common Mistakes

Mistake: Mixing up the signs or coefficients in the triple-angle formulas, such as writing sin⁡3θ=3sin⁡θ+4sin⁡3θ\sin 3\theta = 3\sin\theta + 4\sin^3\theta.
Correction: The correct formula is sin⁡3θ=3sin⁡θ−4sin⁡3θ\sin 3\theta = 3\sin\theta - 4\sin^3\theta. Derive it from sin⁡(2θ+θ)\sin(2\theta + \theta) to verify the signs.

Related Terms