Chapter 7.5, Problem 22E

To determine

To find: the principal value of x in degrees.

Answer to Problem 22E

  $90°$

Explanation of Solution

Given information:

  $\sin x=1+{\cos }^{2}x$

Calculation:

The principal value of x for the given equation as follows:

  $\begin{array}{c}\sin x=1+{\cos }^{2}x\\ \sin x=1+\left(1-{\sin }^{2}x\right)\left[\text{Using Pythagorean identity}\right]\\ \sin x=2-{\sin }^{2}x\\ {\sin }^{2}x+\sin x-2=0\left[\begin{array}{l}{\text{Add sin}}^{2}x\text{ on each side and }\\ \text{then subtract 2 from each side}\end{array}\right]\\ \left(\sin x-1\right)\left(\sin x+2\right)=0\left[\text{Factors}\right]\\ \sin x=1\text{ and }\sin x=-2\left[\text{Solving the factors}\right]\\ x=90°\text{ and }x=\text{no solution }\left[\because \text{ Range of sine is}-\text{1}\le \text{y}\le \text{1}\right]\\ x=90°\end{array}$

Thus the principal value of x is $90°$ .