Trigonometric Equations - R method

The \(R\) Method is a technique used to solve trigonometric equations of the form

\[ a\sin x \pm b\cos x = c \]

Example

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

Our goal is to solve for \(R\) and \(\alpha\) in the following equation:

\[\begin{align} 3\sin x + 4 \cos x &\equiv R\sin (x+\alpha) \\ & = R\sin x \cos \alpha + R \cos x \sin \alpha\\\\ \Rightarrow R\cos \alpha &= 3,\quad R\sin \alpha = 4. \end{align}\]

Taking their ratio leads to

\[\displaystyle \tan \alpha = \frac{4}{3}\implies \alpha = \arctan \frac{4}{3} = 53^\circ 8'.\]

Substituting the value of \(\alpha\) into the previous equation gives

\[R\cos53^\circ 8' = 3\implies R = 5.\]

Finally,

\[3\sin x + 4 \cos x \equiv 5\sin (x+53^\circ 8').\ _\square\]

Generalization

Taking the most suitable form of trigonometric addition formula will save you a lot of work. The equations below state the best \(R\) method in different cases.

\[a\sin x + b \cos x \equiv R\sin (x+\alpha)\] \[a\sin x - b \cos x \equiv R\sin (x-\alpha)\] \[a\cos x + b \sin x \equiv R\cos (x-\alpha)\] \[a\cos x - b \sin x \equiv R\cos (x+\alpha)\]

Problem Solving

\(R\) method can be very effective when solving trigonometric equations and inequalities.

Solve the equation \(3\sin x + 4 \cos x = 1\) for \(0\leq x \leq 180^\circ.\)

Applying the \(R\) method, we have

\[\begin{align} 3 \sin x + 4 \cos x &= 5\sin (x+53^\circ 8') \\ 5\sin (x+53^\circ 8') &= 1 \\ \sin (x+53^\circ 8') &= \frac{1}{5} \\ x+53^\circ 8' &= 168^\circ 28' \\ x &= 115^\circ 20'.\ _\square \end{align}\]

What is the maximum value of \(3\sin x + 4 \cos x?\)

Since the maximum value of the sine function is \(1\), we have

\[\begin{align} 3 \sin x + 4 \cos x &= 5\sin (x+53^\circ 8') \\ &\leq 5.\ _\square \end{align}\]

Contents