Sum Formulas for Sine and Cosine

Theorem

Sine of x plus Cosine of x: Sine Form

$\sin x + \cos x = \sqrt 2 \sin \left({x + \dfrac \pi 4}\right)$

Sine of x plus Cosine of x: Cosine Form

$\sin x + \cos x = \sqrt 2 \, \map \cos {x - \dfrac \pi 4}$

Sine of x minus Cosine of x: Sine Form

$\sin x - \cos x = \sqrt 2 \map \sin {x - \dfrac \pi 4}$

Sine of x minus Cosine of x: Cosine Form

$\sin x - \cos x = \sqrt 2 \, \map \cos {x - \dfrac {3 \pi} 4}$

Cosine of x minus Sine of x: Sine Form

$\cos x - \sin x = \sqrt 2 \, \map \sin {x + \dfrac {3 \pi} 4}$

Cosine of x minus Sine of x: Cosine Form

$\cos x - \sin x = \sqrt 2 \, \map \cos {x + \dfrac \pi 4}$

Multiple of Sine plus Multiple of Cosine

Cosine Form

$p \sin x + q \cos x = \sqrt {p^2 + q^2} \map \cos {x + \arctan \dfrac {-p} q}$

Sine Form

$p \sin x + q \cos x = \sqrt {p^2 + q^2} \map \sin {x + \arctan \dfrac q p}$