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Angle addition formulas

Trigonometric ratios & functions

Japanese school year: Math II

What you learn

Learn addition and subtraction formulas to compute trigonometric values of the sum and difference of two angles. These formulas are fundamental for simplifying trigonometric expressions and analyzing wave patterns. Prior familiarity with coordinates on the unit circle will assist in understanding their derivations and applications.

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Key points

This formula finds the sine of the sum or difference of two angles α\alpha and β\beta. The signs ±\pm match in the same order on both sides.

sin⁡(α±β)=sin⁡αcos⁡β±cos⁡αsin⁡β\sin(\alpha \pm \beta) = \sin\alpha\cos\beta \pm \cos\alpha\sin\beta

This formula finds the cosine of the sum or difference of two angles α\alpha and β\beta. Notice that the signs invert between the left and right sides.

cos⁡(α±β)=cos⁡αcos⁡β∓sin⁡αsin⁡β\cos(\alpha \pm \beta) = \cos\alpha\cos\beta \mp \sin\alpha\sin\beta

This formula finds the tangent of the sum or difference of two angles α\alpha and β\beta. The numerator keeps the same sign, while the denominator inverts it.

tan⁡(α±β)=tan⁡α±tan⁡β1∓tan⁡αtan⁡β\tan(\alpha \pm \beta) = \frac{\tan\alpha \pm \tan\beta}{1 \mp \tan\alpha\tan\beta}

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