Chapter 5, Problem 54RE

To determine

To find all the solutions of the given equation in the interval $\left[0,2\pi \right)$ . Check the result graphically.

Answer to Problem 54RE

  $x=0$

Explanation of Solution

Given:

  ${\cos }^{2}x+\sin x=1$

Concept used:

Pythagorean identity:

  ${\sin }^{2}u+{\cos }^{2}u=1$

Calculation:

Consider the given equation and solve as shown:

  $\begin{array}{l}{\sin }^{2}x+2\cos x=2\\ \Rightarrow 1-{\cos }^{2}x+2\cos x=2\left[\text{Pythagorean}\text{identity}\right]\\ \Rightarrow -{\cos }^{2}x+2\cos x=1\\ \Rightarrow {\cos }^{2}x-2\cos x+1=0\\ \Rightarrow {\left(\cos x-1\right)}^2=0\\ \Rightarrow \cos x-1=0\\ \Rightarrow \cos x=1\\ \Rightarrow x=0\end{array}$

Now, check the result using graphing utility.

  

Hence, the solution points are $\left(0,2\right)$ .