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:

  $\frac{1}{8}(3-4\cos x+\cos 2x)$

Explanation of Solution

Given information:

The following expression:

  ${\sin }^4\frac{x}{2}$

Formula used:

The following power reduction formula

  $\begin{array}{l}{\sin }^{2}u=\frac{1-\cos 2u}{2}\\ {\cos }^{2}u=\frac{1+\cos 2u}{2}\end{array}$

The following expansion:

  ${(a-b)}^2=a^2-2ab+b^2$

Calculation:

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

  $\begin{array}{l}{\sin }^4\frac{x}{2}\text{=}{\left({\text{sin}}^2\frac{x}{2}\right)}^2\text{ [Property of exponents]}\\ ={\left(\frac{1-\cos x}{2}\right)}^2\text{ [Power reduction formula]}\\ =\frac{1}{4}\left(1-2\cos x+{\cos }^{2}x\right)\text{ [Expand]}\\ =\frac{1}{4}\left(1-2\cos x+\frac{1+\cos 2x}{2}\right)\text{ [Power reduction formula]}\\ =\frac{1}{4}\left(1-2\cos x+\frac{1}{2}+\frac{1}{2}\cos 2x\right)\text{ [Seperate fractions]}\\ =\frac{1}{4}\left(\frac{1}{2}(2-4\cos x+1+\cos 2x)\right)\text{ [Factor]}\\ =\frac{1}{8}(2-4\cos x+1+\cos 2x)\text{ [Multiply]}\\ =\frac{1}{8}(3-4\cos x+\cos 2x)\text{ [Add like terms]}\\ \end{array}$

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 ${\text{Y}}_{\text{1}}={\sin }^4\frac{x}{2}$ and ${\text{Y}}_2=\frac{1}{8}(3-4\cos x+\cos 2x)$ 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