# 💯 R Formula | Ingel Soong

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# R Formula

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The R formula or R method can be used to convert the expression $a\, sin \,\theta + b\, cos \,\theta$ into a single sine or cosine term.

## Sine Formula $For$ $a, b, \theta, \in \mathbb{R},$ $a\, sin \,\theta + b\, cos \,\theta = R \, sin \,(\theta + \alpha)$ $where$ $R = sqrt \,(a^2 + b^2), tan \,\alpha = \frac{b}{a}$

## Cosine Formula $For$ $\,\, a, b, \theta, \in \mathbb{R},$ $a\, sin \,\theta + b\, cos \,\theta = R \, cos \,(\theta - \alpha)$ $where$ $\,\, R = sqrt \,(a^2 + b^2), tan \,\alpha = \frac{a}{b}$

## Proof (Sine Formula) $Let$ $\,\, a = R \, cos \, \alpha \,\, and \,\, b = R \, sin \, \alpha$ $Therefore$ $\,\, a^{2} + b^{2} = R^{2} \, (cos ^{2} \alpha + sin ^ {2} \alpha)$ $= R^{2}$ $tan \, \alpha = \frac {sin \, \alpha} {cos \, \alpha}$ $= \frac{b}{a}$ $a \, sin \, \theta + b \, cos \, \theta = R \, sin \, \theta \, cos \, \alpha + R \, cos \, \theta \, sin \alpha$ $= R \, sin \,(\theta + \alpha)$

## Proof (Cosine Formula) $Let$ $\,\, a = R \, sin \, \alpha \,\, and \,\, b = R \, cos \, \alpha$ $Therefore$ $\,\, a^{2} + b^{2} = R^{2} \, (sin ^{2} \alpha + cos ^ {2} \alpha)$ $= R^{2}$ $tan \, \alpha = \frac {sin \, \alpha} {cos \, \alpha}$ $= \frac{a}{b}$ $a \, sin \, \theta + b \, cos \, \theta = R \, cos \, \theta \, cos \, \alpha + R \, sin \, \theta \, sin \alpha$ $= R \, cos \,(\theta - \alpha)$