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Double- & half-angle formulas

Trigonometric ratios & functions

Japanese school year: Math II

What you learn

Learn how to derive and apply double-angle and half-angle formulas from addition theorems. These formulas are vital when solving trigonometric equations and reducing powers in calculus calculations. A firm understanding of fundamental addition formulas will allow you to work through these problems with confidence.

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Key points

This formula expresses sin⁡2θ\sin 2\theta using the original angle θ\theta. It is used to match angles to θ\theta across an expression, or to simplify the product sin⁡θcos⁡θ\sin\theta\cos\theta into a single term.

sin⁡2θ=2sin⁡θcos⁡θ\sin 2\theta = 2\sin\theta\cos\theta

This formula expresses cos⁡2θ\cos 2\theta in terms of the original angle θ\theta. Depending on your calculation, you can choose to rewrite it using only cos⁡θ\cos\theta or only sin⁡θ\sin\theta.

cos⁡2θ=cos⁡2θ−sin⁡2θ=2cos⁡2θ−1=1−2sin⁡2θ\cos 2\theta = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta

This formula calculates the squared sine of a half-angle θ2\frac{\theta}{2} using cos⁡θ\cos\theta. It is used to find values for half-angles or to reduce the power of sin⁡\sin from squared to linear.

sin⁡2θ2=1−cos⁡θ2\sin^2\frac{\theta}{2} = \frac{1 - \cos\theta}{2}

This formula calculates the squared cosine of a half-angle θ2\frac{\theta}{2} using cos⁡θ\cos\theta. It is used to find values for half-angles or to reduce the power of cos⁡\cos from squared to linear.

cos⁡2θ2=1+cos⁡θ2\cos^2\frac{\theta}{2} = \frac{1 + \cos\theta}{2}

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