Chapter 7.4, Problem 43E

To determine

To find:The solution of the equation $3{\sin }^2\theta -7\sin \theta +2=0$ .

Answer to Problem 43E

The solutions of the equation $3{\sin }^2\theta -7\sin \theta +2=0$ are $\theta =0.34+2k\pi$ and $\theta =2.80+2k\pi$ .

Explanation of Solution

Given information:

The given equation is $3{\sin }^2\theta -7\sin \theta +2=0$ .

Calculation:

Simplify the given equation.

  $\begin{array}{c}3{\sin }^2\theta -7\sin \theta +2=0\\ 3{\sin }^2\theta -6\sin \theta -\sin \theta +2=0\\ \left(3\sin \theta -1\right)\left(\sin \theta -2\right)=0\end{array}$

First substitute the factor $3\sin \theta -1=0$ .

Simplify the equation

  $\begin{array}{c}3\sin \theta -1=0\\ 3\sin \theta =1\\ \sin \theta =\frac{1}{3}\end{array}$

As sine has period $2\pi$ . So first find the solutions in $\left[0,2\pi \right)$ . The function sine is negative in third and fourth quadrants.

Therefore, the solution to the equation in $\left[0,2\pi \right)$ are

  $\begin{array}{c}\theta ={\sin }^{-1}\left(\frac{1}{3}\right)\\ \approx 0.34,2.80\end{array}$

As sine repeats its values after every $2\pi$ units.

So, the general solution of the equation is given by

  $\theta =0.34+2k\pi ,2.80+2k\pi$

Therefore, the solutions of the equation $3\sin \theta -1=0$ are $\theta =0.34+2k\pi$ and $\theta =2.80+2k\pi$ .

Now substitute the factor $\sin \theta -2=0$ .

As the value of the function sine is always between $-1$ to $1$ . Therefore, $\sin \theta -2=0$ is not possible for any value of $\theta$ .

Therefore, the solutions of the equation $3{\sin }^2\theta -7\sin \theta +2=0$ are $\theta =0.34+2k\pi$ and $\theta =2.80+2k\pi$ .