Chapter 7.4, Problem 11E

To determine

To find: The solution of the equation $\sin \theta =-0.45$ .

Answer to Problem 11E

The solutions of the equation $\sin \theta =-0.45$ are $\theta =3.61+2k\pi$ and $\theta =-0.47+2k\pi$ .

Explanation of Solution

Given information:

The given equation is $\sin \theta =-0.45$ .

Calculation:

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

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

  $\begin{array}{c}\theta ={\sin }^{-1}\left(-0.45\right)\\ \approx 3.61,-0.47\end{array}$

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

So, the general solution of the equation is given by

  $\theta =3.61+2k\pi ,-0.47+2k\pi$ .

Therefore, the solutions of the equation $\sin \theta =-0.45$ are $\theta =3.61+2k\pi$ and $\theta =-0.47+2k\pi$ .