Chapter 5.5, Problem 44E

To determine

To write: The given expression in terms of the first power of cosine and verify the answer using a graphing utility.

Answer to Problem 44E

The given expression in terms of the first power of cosine can be written as follows:

  $\frac{1}{2}(1-\cos x)$

Explanation of Solution

Given information:

The following expression:

  ${\sin }^2\frac{x}{2}$

Formula used:

The following power reduction formula

  ${\sin }^{2}u=\frac{1-\cos 2u}{2}$

Calculation:

The given expression can be written in terms of the first power of cosine as follows:

  $\begin{array}{l}{\sin }^2\frac{x}{2}=\frac{1-\cos 2\left(\frac{x}{2}\right)}{2}\text{ [Power reducing formula]}\\ =\frac{1-\cos x}{2}\text{ [Multiply]}\\ =\frac{1}{2}-\frac{1}{2}\cos x\text{ [Seperate the fractions]}\\ =\frac{1}{2}(1-\cos x)\text{ [Factor]}\end{array}$

The solution is verified graphically as follows:

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 expressions ${\text{Y}}_{\text{1}}={\sin }^2\frac{x}{2}$ and ${\text{Y}}_2=\frac{1}{2}(1-\cos x)$ which is required to graph.

Step 4: Press GRAPH button to graph the function.

The graph is obtained as:

  

Interpretation:

Since both the curves coincide, the solution is verified