Trigonometric Equation II

Problem 205: Solve the equation $\cos(3\theta) = \frac{1}{2}$ .

Solution: We will use arccosine to solve this problem. Notice that

(1) $\begin{equation*} \cos^{-1} \bigg(\frac{1}{2} \bigg) = \frac{\pi}{3} \end{equation*}$

since $\cos(\pi / 3)=1/2$ . This means that

(2) $\begin{equation*} 3 \theta = \frac{\pi}{3} \quad \text{or} \quad -3\theta = - \frac{\pi}{3} \quad \text{or } \quad 3 \theta = \pm \frac{\pi}{3}. \end{equation*}$

But these are not the only solutions. In general we have,

(3) $\begin{align*}3 \theta & = \pm \frac{\pi}{3} + 2k \pi \\ \theta & = \pm \frac{\pi}{9} + \frac{2}{3}\pi k, \quad \text{for} \ k = 0, \pm 1, \pm 2, \dots . \end{align*}$

Hence,

(4) $\begin{equation*} \theta & = \pm \frac{\pi}{9} + \frac{2}{3}\pi k, \quad \text{for} \ k = 0, \pm 1, \pm 2, \dots.\end{equation*}$

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