Chapter 7.5, Problem 39E

To determine

To find: The simplified form of the equation $\cos \theta \cos 3\theta -\sin \theta \sin 3\theta =0$ and the solution of given equation in the interval $\left[0,2\pi \right]$

Answer to Problem 39E

The simplified form of the equation $\cos \theta \cos 3\theta -\sin \theta \sin 3\theta =0$ is $\cos 4\theta =0$ and the solution for this equation in the range $\left[0,2\pi \right]$ is $\frac{\pi }{8},\frac{3\pi }{8},\frac{5\pi }{8},\frac{7\pi }{8},\frac{9\pi }{8},\frac{11\pi }{8},\frac{13\pi }{8}$ and $\frac{15\pi }{8}$ .

Explanation of Solution

Given information:

The equation is $\cos \theta \cos 3\theta -\sin \theta \sin 3\theta =0$ .

Calculation:

Write the addition formula.

  $\cos \left(a+b\right)=\cos a\cos b-\sin a\sin b$

Simplify given equation using addition formula.

  $\begin{array}{c}\cos \theta \cos 3\theta -\sin \theta \sin 3\theta =0\\ \cos \left(\theta +3\theta \right)=0\\ \cos 4\theta =0\\ 4\theta =\frac{\pi }{2}+k\end{array}$

  $\theta =\frac{\pi }{8}+\frac{\pi k}{4}$

In the above equation $k$ is any integer.

Calculate all the values in the range $\left[0,2\pi \right]$ .

Calculate the value for $k$ =0.

  $\begin{array}{c}\theta =\frac{\pi }{8}+\frac{\pi \times 0}{4}\\ =\frac{\pi }{8}\end{array}$

Calculate the value for $k=1$ .

  $\begin{array}{c}\theta =\frac{\pi }{8}+\frac{\pi \times 1}{4}\\ =\frac{\pi }{8}+\frac{\pi }{4}\\ =\frac{3\pi }{8}\end{array}$

Calculate the value for $k=2$ .

  $\begin{array}{c}\theta =\frac{\pi }{8}+\frac{\pi \times 2}{4}\\ =\frac{\pi }{8}+\frac{\pi }{2}\\ =\frac{5\pi }{8}\end{array}$

Calculate the value for $k=3$ .

  $\begin{array}{c}\theta =\frac{\pi }{8}+\frac{\pi \times 3}{4}\\ =\frac{\pi }{8}+\frac{3\pi }{4}\\ =\frac{7\pi }{8}\end{array}$

Calculate the value for $k=4$ .

  $\begin{array}{c}\theta =\frac{\pi }{8}+\frac{\pi \times 4}{4}\\ =\frac{\pi }{8}+\pi \\ =\frac{9\pi }{8}\end{array}$

Calculate the value for $k=5$ .

  $\begin{array}{c}\theta =\frac{\pi }{8}+\frac{\pi \times 5}{4}\\ =\frac{\pi }{8}+\frac{5\pi }{4}\\ =\frac{11\pi }{8}\end{array}$

Calculate the value for $k=6$ .

  $\begin{array}{c}\theta =\frac{\pi }{8}+\frac{\pi \times 6}{4}\\ =\frac{\pi }{8}+\frac{3\pi }{2}\\ =\frac{13\pi }{8}\end{array}$

Calculate the value for $k=6$ .

  $\begin{array}{c}\theta =\frac{\pi }{8}+\frac{\pi \times 7}{4}\\ =\frac{\pi }{8}+\frac{7\pi }{4}\\ =\frac{15\pi }{8}\end{array}$

Therefore, the simplified form of the equation $\cos \theta \cos 3\theta -\sin \theta \sin 3\theta =0$ is $\cos 4\theta =0$ and the solution for this equation in the range $\left[0,2\pi \right]$ is $\frac{\pi }{8},\frac{3\pi }{8},\frac{5\pi }{8},\frac{7\pi }{8},\frac{9\pi }{8},\frac{11\pi }{8},\frac{13\pi }{8}$ and $\frac{15\pi }{8}$ .