Chapter 5.5, Problem 127E

(a)

To determine

To write: The given function in terms of first power of cosine.

Answer to Problem 127E

The given function in terms of first power of cosine can be written as:

  $\frac{1}{4}(3+\cos 4x)$

Explanation of Solution

Given information:

The following function:

  $f(x)={\sin }^{4}x+{\cos }^{4}x$

Formula used:

The following power reduction formula:

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

The following expansion formula:

  ${(a+b)}^2=a^2+2ab+b^2$

  ${(a-b)}^2=a^2-2ab+b^2$

Calculation:

The given function in terms of first power of cosine can be written as:

  $\begin{array}{l}f(x)={\sin }^{4}x+{\cos }^{4}x\text{[Given]}\\ ={\sin }^{2}x.{\sin }^{2}x+{\cos }^{2}x.{\cos }^{2}x[\text{Property of exponents]}\\ ={\left(\frac{1-\cos 2x}{2}\right)}^2+{\left(\frac{1+\cos 2x}{2}\right)}^2\text{[Power reduction formula]}\\ =\left(\frac{1-2\cos 2x+{\cos }^{2}2x}{2}\right)+\left(\frac{1+2\cos 2x+{\cos }^{2}2x}{2}\right)\text{[Expansion formula]}\\ =\frac{2+2{\cos }^{2}2x}{4}\text{[Add]}\\ =\frac{2(1+{\cos }^{2}2x)}{4}[\text{Factor]}\\ =\frac{1+{\cos }^{2}2x}{2}[\text{Simplify]}\\ =\frac{1+\frac{1+\cos 4x}{2}}{2}\text{[Power reduction formula]}\\ =\frac{3+\cos 4x}{4}\text{[Simplify]}\\ =\frac{1}{4}(3+\cos 4x)\text{[Facctor]}\end{array}$

(b)

To determine

To write: The given function in another way.

Answer to Problem 127E

The given function in another way can be written as:

  $\frac{1+{\cos }^{2}2x}{2}$

Explanation of Solution

Given information:

The following function:

  $f(x)={\sin }^{4}x+{\cos }^{4}x$

Formula used:

The following power reduction formula:

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

The following expansion formula:

  ${(a+b)}^2=a^2+2ab+b^2$

  ${(a-b)}^2=a^2-2ab+b^2$

Calculation:

The given function in another way can be written as:

  $\begin{array}{l}f(x)={\sin }^{4}x+{\cos }^{4}x\text{[Given]}\\ ={\sin }^{2}x.{\sin }^{2}x+{\cos }^{2}x.{\cos }^{2}x[\text{Property of exponents]}\\ ={\left(\frac{1-\cos 2x}{2}\right)}^2+{\left(\frac{1+\cos 2x}{2}\right)}^2\text{[Power reduction formula]}\\ =\left(\frac{1-2\cos 2x+{\cos }^{2}2x}{2}\right)+\left(\frac{1+2\cos 2x+{\cos }^{2}2x}{2}\right)\text{[Expansion formula]}\\ =\frac{2+2{\cos }^{2}2x}{4}\text{[Add]}\\ =\frac{2(1+{\cos }^{2}2x)}{4}[\text{Factor]}\\ =\frac{1+{\cos }^{2}2x}{2}[\text{Simplify]}\end{array}$

This can be 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 expressions ${\text{Y}}_1={\sin }^{4}x+{\cos }^{4}x$ and ${\text{Y}}_2=\frac{1+{\cos }^{2}2x}{2}$ 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 equation is verified.

(c)

To determine

To write: The function after adding a trigonometric term so that it becomes a perfect square trinomial.

Answer to Problem 127E

The function after adding a trigonometric term so that it becomes a perfect square trinomial.

can be written as:

  $1-2{\sin }^{2}x{\cos }^{2}x$

Explanation of Solution

Given information:

The following function:

  $f(x)={\sin }^{4}x+{\cos }^{4}x$

Formula used:

The following expansion formula:

  ${(a+b)}^2=a^2+2ab+b^2$

The following Pythagorean identity:

  ${\sin }^{2}x+{\cos }^{2}x=1$

Calculation:

The given function can be written as a perfect square trinomial as follows:

  $\begin{array}{l}f(x)={\sin }^{4}x+{\cos }^{4}x\text{[Given]}\\ ={\sin }^{4}x+2{\sin }^{2}x{\cos }^{2}x+{\cos }^{4}x-2{\sin }^{2}x{\cos }^{2}x[\text{Add and subtract a trigonometric term]}\\ ={\left({\sin }^{2}x+{\cos }^{2}x\right)}^2-2{\sin }^{2}x{\cos }^{2}x\text{[Expansion formula]}\\ =1-2{\sin }^{2}x{\cos }^{2}x\text{[Pythagorean identity]}\\ \end{array}$

This can be 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 expressions ${\text{Y}}_1={\sin }^{4}x+{\cos }^{4}x$ and ${\text{Y}}_2=1-2{\sin }^{2}x{\cos }^{2}x$ 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 equation is verified.

(d)

To determine

To write: The result in part (c) as sine of double angle.

Answer to Problem 127E

The result in part (c) as sine of double anglecan be written as:

  $1-\frac{1}{2}\sin 2x$

Explanation of Solution

Given information:

The following function:

  $f(x)={\sin }^{4}x+{\cos }^{4}x$

Formula used:

The following double angle formula:

  $\sin 2u=2\sin u\cos u$

Calculation:

The result in part (c) as sine of double angle can be written as follows:

  $\begin{array}{l}f(x)=1-2{\sin }^{2}x{\cos }^{2}x\text{ [Calculated]}\\ \text{ =1}-\frac{1}{2}(4{\sin }^{2}x{\cos }^{2}x)\text{ [Factor]}\\ $ =1-\frac{1}{2}{(2\sin x\cos x)}^2\text{ [Property of exponents]}$=1-\frac{1}{2}{\sin }^{2}2x\text{ [Double angle formula]}\\ \end{array}$

This can be 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 expressions ${\text{Y}}_1=1-2{\sin }^{2}x{\cos }^{2}x$ and ${\text{Y}}_2=1-\frac{1}{2}{\sin }^{2}2x$ 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 equation is verified.

(e)

To determine

To explain: The given statement.

Answer to Problem 127E

No, it does not specifically mean that one of you is wrong.

Explanation of Solution

Given information:

The following statement:

When you rewrite a trigonometric expression, your result may not be the same as a friend. Does this mean that one of you is wrong?

If a friend’s answer is different than yours, it does not specifically mean that one of you is wrong. As seen in the different parts of this sum, a function can be written in many different ways. To verify if your answer is correct, the graphs of the two functions must coincide.