For $a,b>0$, take $R>0$ and acute $\alpha$. $a\sin\theta\pm b\cos\theta=R\sin(\theta\pm\alpha)$ $a\cos\theta\pm b\sin\theta=R\cos(\theta\mp\alpha)$ $R=\sqrt{a^2+b^2}\qquad\tan\alpha=\dfrac ba$ Derivation: expand and compare coefficients StepWorking Match $\sin\theta$ and $\cos\theta$ $①$² + $②$² $②$ ÷ $①$ Maximum and minimum The unrestricted range is $[-R,R]$. For a restricted interval, check that the required angles are included. Single-function formMaximumMinimum $R\sin(\theta+\alpha)$ $R\cos(\theta-\alpha)$ Extremal angles repeat every full revolution. Match the requested sine or cosine form before identifying a and b. Example Express $5\sin\theta+2\cos\theta$ as $R\sin(\theta+\alpha)$, with $\alpha$ in radians. $R=\sqrt{5^2+2^2}=\sqrt{29},\qquad\alpha=\tan^{-1}\!\left(\dfrac25\right)\approx0.381\text{ rad}$ $5\sin\theta+2\cos\theta=\sqrt{29}\sin(\theta+\alpha)$

R-Formulae

For $a,b>0$, take $R>0$ and acute $\alpha$.

$a\sin\theta\pm b\cos\theta=R\sin(\theta\pm\alpha)$

$a\cos\theta\pm b\sin\theta=R\cos(\theta\mp\alpha)$

$R=\sqrt{a^2+b^2}\qquad\tan\alpha=\dfrac ba$

Derivation: expand and compare coefficients

Maximum and minimum

The unrestricted range is $[-R,R]$. For a restricted interval, check that the required angles are included.

Single-function form	Maximum	Minimum
$R\sin(\theta+\alpha)$
$R\cos(\theta-\alpha)$

Extremal angles repeat every full revolution. Match the requested sine or cosine form before identifying a and b.

Example

Express $5\sin\theta+2\cos\theta$ as $R\sin(\theta+\alpha)$, with $\alpha$ in radians.

$R=\sqrt{5^2+2^2}=\sqrt{29},\qquad\alpha=\tan^{-1}\!\left(\dfrac25\right)\approx0.381\text{ rad}$

$5\sin\theta+2\cos\theta=\sqrt{29}\sin(\theta+\alpha)$

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