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)

 

 

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