Chapter 5.4, Problem 88E

(a)

To determine

To write: The given expressions in the given form.

Answer to Problem 88E

The given expression can be written as:

  $\sqrt{2}\sin (2\theta +\pi /4)$

Explanation of Solution

Given information:

The following expression:

  $\sin 2\theta +\cos 2\theta$

The following form:

  $\sqrt{a^2+b^2}\sin (B\theta +C)$

The following formulas in Exercises 85 and 86:

  $\begin{array}{l}a\sin B\theta +b\cos B\theta =\sqrt{a^2+b^2}\sin (B\theta +C),\text{ where }C=\arctan (b/a)\\ a\sin B\theta +b\cos B\theta =\sqrt{a^2+b^2}\cos (B\theta -C),\text{ where }C=\arctan (b/a)\\ \end{array}$

Calculation:

To write the given expression in the given form, compare $\sin 2\theta +\cos 2\theta$ with $a\sin B\theta +b\cos B\theta$

On comparing,

  $\begin{array}{l}a=1\\ b=1\\ B=2\\ C=\arctan (b/a)\text{ [Given]}\\ \text{ = arctan(1/1) [Substitute]}\\ \text{ =}\pi /4\text{ [Find value]}\\ \end{array}$

Now, substitute these values in $\sqrt{a^2+b^2}\sin (B\theta +C)$ ,

  $\begin{array}{l}\sqrt{a^2+b^2}\sin (B\theta +C)\text{[Given]}\\ =\sqrt{1^2+1^2}\sin (2\theta +\pi /4)\text{[Substitute]}\\ =\sqrt{2}\sin (2\theta +\pi /4)\text{[Simplify]}\end{array}$

To verify this, plot the graph of the given expression and the obtained expression in the same window:

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 ${\text{Y}}_1=\sin 2\theta +\cos 2\theta$ and ${\text{Y}}_2=\sqrt{2}\sin (2\theta +\pi /4)$ 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 expressions are equal. Hence, verified.

(b)

To determine

To write: The given expressions in the given form.

Answer to Problem 88E

The given expression can be written as:

  $\sqrt{2}\cos (2\theta -\pi /4)$

Explanation of Solution

Given information:

The following expression:

  $\sin 2\theta +\cos 2\theta$

The following form:

  $\sqrt{a^2+b^2}\cos (B\theta -C)$

The following formulas in Exercises 85 and 86:

  $\begin{array}{l}a\sin B\theta +b\cos B\theta =\sqrt{a^2+b^2}\sin (B\theta +C),\text{ where }C=\arctan (b/a)\\ a\sin B\theta +b\cos B\theta =\sqrt{a^2+b^2}\cos (B\theta -C),\text{ where }C=\arctan (b/a)\\ \end{array}$

Calculation:

To write the given expression in the given form, compare $\sin 2\theta +\cos 2\theta$ with $a\sin B\theta +b\cos B\theta$

On comparing,

  $\begin{array}{l}a=1\\ b=1\\ B=2\\ C=\arctan (b/a)\text{ [Given]}\\ \text{ = arctan(1/1) [Substitute]}\\ \text{ =}\pi /4\text{ [Find value]}\\ \end{array}$

Now, substitute these values in $\sqrt{a^2+b^2}\cos (B\theta -C)$ ,

  $\begin{array}{l}\sqrt{a^2+b^2}\cos (B\theta -C)\text{[Given]}\\ =\sqrt{1^2+1^2}\cos (2\theta -\pi /4)\text{[Substitute]}\\ =\sqrt{2}\cos (2\theta -\pi /4)\text{[Simplify]}\end{array}$

To verify this, plot the graph of the given expression and the obtained expression in the same window:

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 ${\text{Y}}_1=\sin 2\theta +\cos 2\theta$ and $\sqrt{2}\cos (2\theta -\pi /4)$ 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 expressions are equal. Hence, verified.