Chapter 12.4, Problem 44WE

To determine

Find the measure of angle $\left(\theta \right)$ to the nearest tenth of degree between $0°\text{ and }360°$ for the function $\sin \theta =-0.5212$ and satisfy the condition $90°<\theta <270°$ .

Answer to Problem 44WE

The angle $\left(\theta \right)$ is equal to $211.4°$ .

Explanation of Solution

Given information: The given function is $\sin \theta =-0.5212$ which satisfy the condition $90°<\theta <270°$ .

Calculation: The value of angle $\left(\theta \right)$ for the given function $\sin \theta =-0.5212$ is evaluated as,

  $\begin{array}{c}\sin \theta =-0.5212\\ \theta ={\sin }^{-1}\left(-0.5212\right)\\ =-31.4°\\ =-31.4°+\left(360°\right)\mathrm{....}\left(\begin{array}{l}\text{Add 360}°\text{ to }\\ \text{make positive}\end{array}\right)\\ =328.6°\end{array}$

Now according to question $\sin \theta =-0.5212$ is negative and the angle $\left(\theta \right)$ obtained is $328.6°$ which is also positive and lies in fourth quadrant but according to question $90°<\theta <270°$ . Therefore angle $\left(\theta \right)$ which satisfies the condition $90°<\theta <270°$ can be obtained as,

  $\begin{array}{c}\theta =328.6°-270°\\ =58.6°\\ =270°-58.6°\text{for 90}°\text{<}\theta <270°\\ =211.4°\text{ for 90}°\text{<}\theta <270°\end{array}$