A calculus of the absurd

7.5 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 7.5).

\begin{align*} R \sin (\theta + \alpha ) &= R \cos (\alpha ) \sin (\theta ) + R \sin (\alpha ) \cos (\theta ) \\ &= \hspace {32pt} 1 \sin (\theta ) + \hspace {32pt} 3 \cos (\theta ) \end{align*}

From here, comparing coefficients gives

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

This means

\begin{align*} & R^2 \cos ^2(\alpha ) + R^2 \sin ^2(\alpha ) = 1 + 3^2 \\ & R^2 (\cos ^2(\alpha ) + sin^2(\alpha ) = 10 \\ & R^2 = 10 \\ & R = \sqrt {10} \end{align*}

as well as that

\begin{align*} & \frac {R\sin (\alpha )}{R\cos (\alpha )} = 3 \\ & \tan (\alpha ) = 3 \\ & \alpha = \arctan (3) \end{align*}

So the solution is

\begin{equation*} sin(\theta ) + 3cos(\theta ) = \sqrt {10} \sin \left (x + \arctan (3) \right ) \end{equation*}

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 functions5656 Explored in the section above..