# 7.5 Writing sums of trig functions as a single trig function

###### Example 7.5.1

Express $3\cos(\theta)+4\sin(\theta)$ in the form $R\sin(\theta+\alpha)$.

Solution: start by applying the angle addition formula for $\sin(\theta)$ (Equation 7.8).

 $\displaystyle R\sin(\theta+\alpha)$ $\displaystyle=R\cos(\alpha)\sin(\theta)+R\sin(\alpha)\cos(\theta)$ $\displaystyle=\hskip 32.0pt1\sin(\theta)+\hskip 32.0pt3\cos(\theta)$

From here, comparing coefficients gives

$\begin{cases}R\cos(\alpha)=1\\ R\sin(\alpha)=3\end{cases}$

This means

 $\displaystyle R^{2}\cos^{2}(\alpha)+R^{2}\sin^{2}(\alpha)=1+3^{2}$ $\displaystyle R^{2}(\cos^{2}(\alpha)+sin^{2}(\alpha)=10$ $\displaystyle R^{2}=10$ $\displaystyle R=\sqrt{10}$

as well as that

 $\displaystyle\frac{R\sin(\alpha)}{R\cos(\alpha)}=3$ $\displaystyle\tan(\alpha)=3$ $\displaystyle\alpha=\arctan(3)$

So the solution is

$sin(\theta)+3cos(\theta)=\sqrt{10}\sin\left(x+\arctan(3)\right)$

Note that this technique is very useful for solving equations of the form $A\cos(\theta)+B\sin(\theta)=c$, as we just rewrite the left hand side as a single trigonometric function, and then use the method for solving such trig functions66 6 Explored in the section above..