Chapter 7.5, Problem 27E

(a)

To determine

To find: The solution of the equation $\sin 2\theta =3\cos 2\theta$ .

Answer to Problem 27E

The solutions of the equation $\sin 2\theta =3\cos 2\theta$ are $\theta =0.62+\frac{k\pi }{2}$ .

Explanation of Solution

Given information:

The given equation is $\sin 2\theta =3\cos 2\theta$ .

Calculation:

Simplify the equation.

  $\begin{array}{c}\sin 2\theta =3\cos 2\theta \\ \frac{\sin 2\theta }{\cos 2\theta }=3\\ \tan 2\theta =3\end{array}$

As tangent has period $\pi$ . So first find the solutions in $\left(-\frac{\pi }{2},\frac{\pi }{2}\right)$ .

Therefore, the solution to the equation in $\left(-\frac{\pi }{2},\frac{\pi }{2}\right)$ are

  $\begin{array}{c}2\theta ={\tan }^{-1}\left(3\right)\\ =1.25\end{array}$

As tangent repeats its values after every $\pi$ units.

So, the general solution of the equation is given by

  $\begin{array}{c}2\theta =1.25+k\pi \\ \theta =0.62+\frac{k\pi }{2}\end{array}$

Therefore, the solutions of the equation $\sin 2\theta =3\cos 2\theta$ are $\theta =0.62+\frac{k\pi }{2}$ .

(b)

To determine

To find: The solution of the equation $\sin 2\theta =3\cos 2\theta$ in $\left[0,2\pi \right)$ .

Answer to Problem 27E

The solution of the equation $\sin 2\theta =3\cos 2\theta$ in $\left[0,2\pi \right)$ is $\theta =0.62,2.19,3.76,5.33$ .

Explanation of Solution

Given information:

The given equation is $\sin 2\theta =3\cos 2\theta$ and the interval is $\left[0,2\pi \right)$ .

Calculation:

According to solution in part(a) the general solution of the equation $\sin 2\theta =3\cos 2\theta$ are $\theta =0.62+\frac{k\pi }{2}$ . The solution of the equation in $\left[0,2\pi \right)$ corresponds to $k=0,1,2,3$ i.e. $\theta =0.62,2.19,3.76,5.33$ .

Therefore, the solution of the equation $\sin 2\theta =3\cos 2\theta$ in $\left[0,2\pi \right)$ is $\theta =0.62,2.19,3.76,5.33$ .