Trigonometric Equations - R method
The R Method is a technique used to solve trigonometric equations in the form
\[ a\sin x \pm b\cos x = c \]
by applying the Addition Formulae.
Contents
Example
Express \(3\sin x + 4 \cos x\) in the form \(R\sin (x+\alpha)\).
Solution: Our goal is to solve for \(R\) and \(\alpha\) in the following equation.
\[\begin{align} 3\sin x + 4 \cos x &\equiv R\sin (x+\alpha) \\ & \equiv R\sin x \cos \alpha + R \cos x \sin \alpha \end{align}\]
\[R\cos \alpha = 3\]
\[R\sin \alpha = 4\]
Taking their ratio leads to
\[\displaystyle \tan \alpha = \frac{4}{3}\]
\[\begin{align} \alpha &= \arctan \frac{4}{3} \\ &= 53^\circ 8' \end{align}\]
Substitute the value of \(\alpha\) to the previous equation
\[R\cos53^\circ 8' = 3\]
\[R = 5\]
Finally,
\[3\sin x + 4 \cos x \equiv 5\sin (x+53^\circ 8')\]
Generalisation
Taking the most suitable form of trigonometric addition formula will save you a lot of work. The equations below states 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 \(0\leq x \leq 180^\circ\)
Solution: Applying the R method, we have
\[\begin{align} 3 \sin x + 4 \cos x &= 5\sin (x+53^\circ 8') \\ 5\sin (x+53^\circ 8') &= 1 \\ \sin (x+53^\circ 8') &= \frac{1}{5} \\ x+53^\circ 8' &= 168^\circ 28' \\ x &= 115^\circ 20' \end{align}\]
What is the maximum value of \(3\sin x + 4 \cos x\) ?
Solution: Since the maximum value of the sine function is \(1\), we have
\[\begin{align} 3 \sin x + 4 \cos x &= 5\sin (x+53^\circ 8') \\ &\leq 5 \end{align}\]