Square

# 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)

$Let$ $\,\, a = R \, sin \, \alpha \,\, and \,\, b = R \, cos \, \alpha$

$Therefore$ $\,\, a^{2} + b^{2} = R^{2} \, (sin ^{2} \alpha + cos ^ {2} \alpha)$

$= R^{2}$

$tan \, \alpha = \frac {sin \, \alpha} {cos \, \alpha}$

$= \frac{a}{b}$

$a \, sin \, \theta + b \, cos \, \theta = R \, cos \, \theta \, cos \, \alpha + R \, sin \, \theta \, sin \alpha$

$= R \, cos \,(\theta - \alpha)$