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Business Mathematics and Statistics · Ch 4 — Trigonometry

Compound Angle Formulae

5

Compound Angle Formulae

A compound angle is an angle expressed as the algebraic sum or difference of two (or more) angles, such as A+BA+B or A−BA-B. Importantly, sin⁡(A+B)\sin(A+B) is not equal to sin⁡A+sin⁡B\sin A + \sin B — the correct expansions are the compound angle formulae:

sin⁡(A+B)=sin⁡Acos⁡B+cos⁡Asin⁡B\sin(A+B) = \sin A\cos B + \cos A\sin B

sin⁡(A−B)=sin⁡Acos⁡B−cos⁡Asin⁡B\sin(A-B) = \sin A\cos B - \cos A\sin B

cos⁡(A+B)=cos⁡Acos⁡B−sin⁡Asin⁡B\cos(A+B) = \cos A\cos B - \sin A\sin B

cos⁡(A−B)=cos⁡Acos⁡B+sin⁡Asin⁡B\cos(A-B) = \cos A\cos B + \sin A\sin B

Dividing the sine formula by the cosine formula (and then dividing numerator and denominator by cos⁡Acos⁡B\cos A\cos B) gives the tangent versions:

tan⁡(A+B)=tan⁡A+tan⁡B1−tan⁡Atan⁡B,tan⁡(A−B)=tan⁡A−tan⁡B1+tan⁡Atan⁡B\tan(A+B) = \frac{\tan A + \tan B}{1 - \tan A\tan B}, \qquad \tan(A-B) = \frac{\tan A - \tan B}{1 + \tan A\tan B} …

Definition 1Sine and Cosine Compound-Angle Formulae

sin⁡(A±B)=sin⁡Acos⁡B±cos⁡Asin⁡B\sin(A\pm B)=\sin A\cos B\pm\cos A\sin B; $\cos(A\pm B)=\cos A\cos B\mp …

Definition 2Tangent Compound-Angle Formula

tan⁡(A±B)=tan⁡A±tan⁡B1∓tan⁡Atan⁡B\tan(A\pm B)=\dfrac{\tan A\pm\tan B}{1\mp\tan A\tan B}, valid whenever all the terms invo …

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