Chapter 7.5, Problem 17E

a.

To determine

To solve : the given equation $\{2\cos 3\theta =1$ .

Answer to Problem 17E

The general solutions of $\theta$ for the equation $\{2\cos 3\theta =1$ are $\theta =\frac{\pi }{9}+\frac{2}{3}n\pi \text{ or }\theta =\frac{5\pi }{9}+\frac{2}{3}n\pi$ .

Explanation of Solution

Given information:

The equation to be solved is $\{2\cos 3\theta =1$ .

Calculation :

  $2\cos 3\theta =1$

Solve the given equation.

  $\cos 3\theta =\frac{1}{2}$

Above equation is satisfied in the interval $\left[0,2\pi \right]$ when $3\theta =\frac{\pi }{3}\text{ or }3\theta =\frac{5\pi }{3}$ .

Since, the period of cosine function is $2\pi$ . Add $2\pi n$ to get the general solution of $\theta$ where $n$ is any integer.

  $\begin{array}{l}3\theta =\frac{\pi }{3}\text{+2n}\pi \text{ or }3\theta =\frac{5\pi }{3}+2n\pi \\ \theta =\frac{\pi }{9}+\frac{2}{3}n\pi \text{ or }\theta =\frac{5\pi }{9}+\frac{2}{3}n\pi \end{array}$

Therefore, the general solutions of $\theta$ for the equation $\{2\cos 3\theta =1$ are

  $\theta =\frac{\pi }{9}+\frac{2}{3}n\pi \text{ or }\theta =\frac{5\pi }{9}+\frac{2}{3}n\pi$ .

b.

To determine

To find: the solutions of the equation $2\cos 3\theta =1$ in the interval $[0,2\pi )$ .

Answer to Problem 17E

The general solutions of equation are $\{2\cos 3\theta =1$ in the interval $[0,2\pi )$ are $\theta =\frac{\pi }{9},\frac{5\pi }{9},\frac{7\pi }{9},\frac{11\pi }{9},\frac{13\pi }{9}\text{ or }\frac{17\pi }{9}$

Explanation of Solution

Given information:

The equation to be solved is $2\cos 3\theta =1$ in the interval $[0,2\pi )$ .

Calculation:

The general solutions of $\theta$ for the equation $\{2\cos 3\theta =1$ are

  $\theta =\frac{\pi }{9}+\frac{2}{3}n\pi \text{ or }\theta =\frac{5\pi }{9}+\frac{2}{3}n\pi$ .

Put $n=0,1,2,\mathrm{3......}$ in such a way that $\theta$ should not exceed $2\pi$ because $\theta \in [0,2\pi )$ .

At $n=0$ :

  $\theta =\frac{\pi }{9}\text{ or }\theta =\frac{5\pi }{9}$

At $n=1$ :

  $\theta =\frac{7\pi }{9}\text{ or }\theta =\frac{11\pi }{9}$

At $n=2$ :

  $\theta =\frac{13\pi }{9}\text{ or }\theta =\frac{17\pi }{9}$

Therefore, the general solutions of equation are $\{2\cos 3\theta =1$ in the interval $[0,2\pi )$ are

  $\theta =\frac{\pi }{9},\frac{5\pi }{9},\frac{7\pi }{9},\frac{11\pi }{9},\frac{13\pi }{9}\text{ or }\frac{17\pi }{9}$ .