Chapter 5.5, Problem 72E

To determine

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

Answer to Problem 72E

The solutions of the given equation in the interval $[0,2\pi )$ are $x=0,\frac{\pi }{2},\frac{3\pi }{2}$ .

Explanation of Solution

Given information:

The following equation:

  $\tan \frac{x}{2}-\sin x=0$

The following interval:

  $[0,2\pi )$

Formula used:

The following half angle formula

  $\tan \frac{u}{2}=\frac{1-\cos u}{\sin u}$

The following Pythagorean relation:

  ${\sin }^{2}x+{\cos }^{2}x=1$

Calculation:

The given equation can be solved as follows:

  $\begin{array}{l}\tan \frac{x}{2}-\sin x=0\text{ [Given]}\\ \frac{1-\cos x}{\sin x}-\sin x=0\text{ [Half angle formula]}\\ \frac{1-\cos -{\sin }^{2}x}{\sin x}=0\text{ [Subtract fractions]}\\ \frac{{\cos }^{2}x-\cos x}{\sin x}=0\text{ [Pythagorean identity]}\\ {\cos }^{2}x-\cos x=0\text{ [Multiply both sides by }\sin x]\\ \cos x(\cos x-1)=0\text{ [Factor]}\\ \cos x=0\cos x-1=0\text{ [Set each factor to 0]}\\ \cos x=0\cos x=1\text{ [Simplify]}\\ x=\arccos (0)x=\arccos (1)\text{ [Inverse trigonometric function]}\\ x=\frac{\pi }{2},\frac{3\pi }{2}x=0,2\pi \text{ [Find value of inverse]}\end{array}$

Among all the solutions of $x$ , $x=0,\frac{\pi }{2},\frac{3\pi }{2}$ lie 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 $\tan \frac{x}{2}-\sin x$ 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 $x=0,\frac{\pi }{2},\frac{3\pi }{2}$ . Hence, verified.