Chapter 5.5, Problem 67E

To determine

To find: The solutions of the given equation in the given interval.

Answer to Problem 67E

The solution of the given equation inthe interval $[0,2\pi )$ is $x=\pi$ .

Explanation of Solution

Given information:

The following equation:

  $\sin \frac{x}{2}+\cos x=0$

The following interval:

  $[0,2\pi )$

Formula used:

The following double angle formula

  $\cos 2u=1-2{\sin }^{2}u$

Calculation:

The given equation can be solved as follows:

  $\begin{array}{l}\sin \frac{x}{2}+\cos x=0\text{ [Given]}\\ \sin \frac{x}{2}+\cos 2\left(\frac{x}{2}\right)=0\text{ [Multiply and divide by 2]}\\ \sin \frac{x}{2}+1-2{\sin }^2\frac{x}{2}=0\text{ [Double angle formula]}\\ 2{\sin }^2\frac{x}{2}-\sin \frac{x}{2}-1=0\text{ [Reverse signs to make first term positive]}\\ \left(2\sin \frac{x}{2}+1\right)\left(\sin \frac{x}{2}-1\right)=0\text{ [Factor]}\\ 2\sin \frac{x}{2}+1=0\sin \frac{x}{2}-1=0\text{ [Set each factor to 0]}\\ \sin \frac{x}{2}=-\frac{1}{2}\text{ sin}\frac{x}{2}=1\text{ [Simplify]}\\ \frac{x}{2}=\arcsin \left(-\frac{1}{2}\right)\frac{x}{2}=\arcsin (1)\text{ [Inverse trigonometric function]}\\ \frac{x}{2}=\frac{7\pi }{6},\frac{11\pi }{6}\frac{x}{2}=\frac{\pi }{2}\text{ [Find values of inverse]}\\ x=\frac{7\pi }{3},\frac{11\pi }{3}x=\pi \text{ [Multiply both sides by 2]}\end{array}$

Among all the solutions of $x$ , only $x=\pi$ lies in the given interval. The solution is verified graphically as

Graph:

Sketch the graph using graphing utility.

Step 1: Press WINDOW button to access the Window editor.

Step 2: Press $\boxed{\text{Y=}}$ button.

Step 3: Enter the expression $\sin \frac{x}{2}+\cos x$ and which is required to graph.

Step 4: Press GRAPH button to graph the function.

The graph is obtained as:

  

Interpretation:

In the interval $[0,2\pi )$ , the value of the equation is $0$ at $x=\pi$ . Hence verified.