Chapter 5.5, Problem 9E

(a)

To determine

To calculate:The exact value of the trigonometric function $\sin \theta$ .

Answer to Problem 9E

The exact value of the trigonometric function $\sin \theta$ is $\frac{3}{5}$ .

Explanation of Solution

Given Information:

The figure is given as follows.

  

Formula Used:

Use the formula $\sin \theta =\frac{\text{opp}}{\text{hyp}}$ .

Calculation:

From the figure, $\text{opp}=3$ and $\text{adj}=4$ .

Apply the Pythagoras theorem to calculate the hypotenuse of the triangle.

  $\begin{array}{c}\text{hyp}=\sqrt{{\left(\text{adj}\right)}^2+{\left(\text{opp}\right)}^2}\\ =\sqrt{{\left(4\right)}^2+{\left(3\right)}^2}\\ =\sqrt{16+9}\\ =\sqrt{25}\\ =5\end{array}$ .

Therefore the value of $\text{hyp}=5$ .

Substitute these values in the formula

  $\begin{array}{c}\sin \theta =\frac{\text{opp}}{\text{hyp}}\\ =\frac{3}{5}\end{array}$ .

Therefore, the exact value of $\sin \theta$ is $\frac{3}{5}$ .

(b)

To determine

To calculate:The exact value of the trigonometric function $\cos \theta$ .

Answer to Problem 9E

The exact value of the trigonometric function $\cos \theta$ is $\frac{4}{5}$ .

Explanation of Solution

Given Information:

The figure is given as follows.

  

Formula Used:

Use the formula $\cos \theta =\frac{\text{adj}}{\text{hyp}}$ .

Calculation:

From the figure, $\text{opp}=3$ and $\text{adj}=4$ .

Apply the Pythagoras theorem to calculate the hypotenuse of the triangle.

  $\begin{array}{c}\text{hyp}=\sqrt{{\left(\text{adj}\right)}^2+{\left(\text{opp}\right)}^2}\\ =\sqrt{{\left(4\right)}^2+{\left(3\right)}^2}\\ =\sqrt{16+9}\\ =\sqrt{25}\\ =5\end{array}$ .

Therefore the value of $\text{hyp}=5$ .

Substitute these values in the formula

  $\begin{array}{c}\cos \theta =\frac{\text{adj}}{\text{hyp}}\\ =\frac{4}{5}\end{array}$ .

Therefore, the exact value of $\cos \theta$ is $\frac{4}{5}$ .

(c)

To determine

To calculate:The exact value of the trigonometric function $\tan \theta$ .

Answer to Problem 9E

The exact value of the trigonometric function $\tan \theta$ is $\frac{3}{4}$ .

Explanation of Solution

Given Information:

The figure is given as follows.

  

Formula Used:

Use the formula $\tan \theta =\frac{\text{opp}}{\text{hyp}}$ .

Calculation:

From the figure, $\text{opp}=3$ and $\text{adj}=4$ .

Apply the Pythagoras theorem to calculate the hypotenuse of the triangle.

  $\begin{array}{c}\text{hyp}=\sqrt{{\left(\text{adj}\right)}^2+{\left(\text{opp}\right)}^2}\\ =\sqrt{{\left(4\right)}^2+{\left(3\right)}^2}\\ =\sqrt{16+9}\\ =\sqrt{25}\\ =5\end{array}$ .

Therefore the value of $\text{hyp}=5$ .

Substitute these values in the formula

  $\begin{array}{c}\tan \theta =\frac{\text{opp}}{\text{adj}}\\ =\frac{3}{4}\end{array}$ .

Therefore, the exact value of $\tan \theta$ is $\frac{3}{4}$ .

(d)

To determine

To calculate:The exact value of the trigonometric function $\sin 2\theta$ .

Answer to Problem 9E

The exact value of the trigonometric function $\sin 2\theta$ is $\frac{24}{25}$ .

Explanation of Solution

Given Information:

The figure is given as follows.

  

Formula Used:

Use the formula $\sin 2\theta =2\sin \theta \cos \theta$ .

Calculation:

From the given figure, $\text{opp}=3$ and $\text{adj}=4$ .

Apply the Pythagoras theorem to calculate the hypotenuse of the triangle.

  $\begin{array}{c}\text{hyp}=\sqrt{{\left(\text{adj}\right)}^2+{\left(\text{opp}\right)}^2}\\ =\sqrt{{\left(4\right)}^2+{\left(3\right)}^2}\\ =\sqrt{16+9}\\ =\sqrt{25}\\ =5\end{array}$ .

Therefore the value of $\text{hyp}=5$ .

Substitute these values in the formula

  $\begin{array}{c}\sin \theta =\frac{\text{opp}}{\text{hyp}}\\ =\frac{3}{5}\end{array}$ .

Therefore, the exact value of $\sin \theta$ is $\frac{3}{5}$ .

Substitute the values in the formula

  $\begin{array}{c}\cos \theta =\frac{\text{adj}}{\text{hyp}}\\ =\frac{4}{5}\end{array}$ .

Therefore the exact value of $\cos \theta$ is $\frac{4}{5}$ .

Substitute both the values in the formula

  $\begin{array}{l}\sin 2\theta =2\sin \theta \cos \theta \\ =2\times \frac{3}{5}\times \frac{4}{5}\\ =\frac{24}{25}\end{array}$ .

Therefore the exact value of $\sin 2\theta$ is $\frac{24}{25}$ .

(e)

To determine

To calculate:The exact value of the trigonometric function $\cos 2\theta$ .

Answer to Problem 9E

The exact value of the trigonometric function $\cos 2\theta$ is $\frac{7}{25}$ .

Explanation of Solution

Given Information:

The figure is given as follows.

  

Formula Used:

Use the formula $\cos 2\theta =1-2{\sin }^2\theta$ .

Calculation:

From the above, $\text{opp}=3$ and $\text{adj}=4$ .

Apply the Pythagoras theorem to calculate the hypotenuse of the triangle.

  $\begin{array}{c}\text{hyp}=\sqrt{{\left(\text{adj}\right)}^2+{\left(\text{opp}\right)}^2}\\ =\sqrt{{\left(4\right)}^2+{\left(3\right)}^2}\\ =\sqrt{16+9}\\ =\sqrt{25}\\ =5\end{array}$ .

Therefore the value of $\text{hyp}=5$ .

Substitute these values in the formula

  $\begin{array}{c}\sin \theta =\frac{\text{opp}}{\text{hyp}}\\ =\frac{3}{5}\end{array}$ .

Therefore, the exact value of $\sin \theta$ is $\frac{3}{5}$ .

Substitute the value in the formula

  $\begin{array}{c}\cos 2\theta =1-2{\sin }^2\theta \\ =1-2{\left(\frac{3}{5}\right)}^2\\ =1-2\left(\frac{9}{25}\right)\\ =1-\frac{18}{25}\\ =\frac{7}{25}\end{array}$ .

Therefore the exact value of $\cos 2\theta$ is $\frac{7}{25}$ .

(f)

To determine

To calculate:The exact value of the trigonometric function $\sec 2\theta$ .

Answer to Problem 9E

The exact value of the trigonometric function $\sec 2\theta$ is $\frac{25}{7}$ .

Explanation of Solution

Given Information:

The figure is given as follows.

  

Formula Used:

Use the formula $\sec 2\theta =\frac{1}{\cos 2\theta }$ .

Calculation:

From the given figure, $\text{opp}=3$ and $\text{adj}=4$ .

Apply the Pythagoras theorem to calculate the hypotenuse of the triangle.

  $\begin{array}{c}\text{hyp}=\sqrt{{\left(\text{adj}\right)}^2+{\left(\text{opp}\right)}^2}\\ =\sqrt{{\left(4\right)}^2+{\left(3\right)}^2}\\ =\sqrt{16+9}\\ =\sqrt{25}\\ =5\end{array}$ .

Therefore the value of $\text{hyp}=5$ .

Substitute these values in the formula

  $\begin{array}{c}\sin \theta =\frac{\text{opp}}{\text{hyp}}\\ =\frac{3}{5}\end{array}$ .

Therefore, the exact value of $\sin \theta$ is $\frac{3}{5}$ .

Substitute the value in the formula

  $\begin{array}{c}\cos 2\theta =1-2{\sin }^2\theta \\ =1-2{\left(\frac{3}{5}\right)}^2\\ =1-2\left(\frac{9}{25}\right)\\ =1-\frac{18}{25}\\ =\frac{7}{25}\end{array}$ .

Therefore the exact value of $\cos 2\theta$ is $\frac{7}{25}$ .

Substitute $\frac{7}{25}$ for $\cos 2\theta$ in the formula

  $\begin{array}{c}\sec 2\theta =\frac{1}{\cos 2\theta }\\ =\frac{1}{\frac{7}{25}}\\ =\frac{25}{7}\end{array}$ .

Therefore the exact value of $\sec 2\theta$ is $\frac{25}{7}$ .

(g)

To determine

To calculate:The exact value of the trigonometric function $\csc 2\theta$ .

Answer to Problem 9E

The exact value of the trigonometric function $\csc 2\theta$ is $\frac{25}{24}$ .

Explanation of Solution

Given Information:

The figure is given as follows.

  

Formula Used:

Use the formula $\csc 2\theta =\frac{1}{\sin 2\theta }$ .

Calculation:

From the given figure, $\text{opp}=3$ and $\text{adj}=4$ .

Apply the Pythagoras theorem to calculate the hypotenuse of the triangle.

  $\begin{array}{c}\text{hyp}=\sqrt{{\left(\text{adj}\right)}^2+{\left(\text{opp}\right)}^2}\\ =\sqrt{{\left(4\right)}^2+{\left(3\right)}^2}\\ =\sqrt{16+9}\\ =\sqrt{25}\\ =5\end{array}$ .

Therefore the value of $\text{hyp}=5$ .

Substitute these values in the formula

  $\begin{array}{c}\sin \theta =\frac{\text{opp}}{\text{hyp}}\\ =\frac{3}{5}\end{array}$ .

Therefore, the exact value of $\sin \theta$ is $\frac{3}{5}$ .

Substitute the values in the formula

  $\begin{array}{c}\cos \theta =\frac{\text{adj}}{\text{hyp}}\\ =\frac{4}{5}\end{array}$ .

Therefore the exact value of $\cos \theta$ is $\frac{4}{5}$ .

Substitute both the values in the formula

  $\begin{array}{l}\sin 2\theta =2\sin \theta \cos \theta \\ =2\times \frac{3}{5}\times \frac{4}{5}\\ =\frac{24}{25}\end{array}$ .

Therefore the exact value of $\sin 2\theta$ is $\frac{24}{25}$ .

Substitute this value in the formula

  $\begin{array}{c}\csc 2\theta =\frac{1}{\sin 2\theta }\\ =\frac{1}{\frac{24}{25}}\\ =\frac{25}{24}\end{array}$ .

Therefore the exact value of $\csc 2\theta$ is $\frac{25}{24}$ .

(h)

To determine

To calculate:The exact value of the trigonometric function $\cot 2\theta$ .

Answer to Problem 9E

The exact value of the trigonometric function $\cot 2\theta$ is $\frac{24}{25}$ .

Explanation of Solution

Given Information:

The figure is given as follows.

  

Formula Used:

Use the formula $\cot 2\theta =\frac{1}{\tan 2\theta }$ , where $\tan 2\theta =\frac{2\tan \theta }{1-{\tan }^2\theta }$ .

Calculation:

From the given figure, $\text{opp}=3$ and $\text{adj}=4$ .

Apply the Pythagoras theorem to calculate the hypotenuse of the triangle.

  $\begin{array}{c}\text{hyp}=\sqrt{{\left(\text{adj}\right)}^2+{\left(\text{opp}\right)}^2}\\ =\sqrt{{\left(4\right)}^2+{\left(3\right)}^2}\\ =\sqrt{16+9}\\ =\sqrt{25}\\ =5\end{array}$ .

Therefore the value of $\text{hyp}=5$ .

Substitute these values in the formula

  $\begin{array}{c}\tan \theta =\frac{\text{opp}}{\text{adj}}\\ =\frac{3}{4}\end{array}$ .

Therefore, the exact value of $\tan \theta$ is $\frac{3}{4}$ .

Substitute the value in the formula

  $\begin{array}{c}\tan 2\theta =\frac{2\tan \theta }{1-{\tan }^2\theta }\\ =\frac{2\times \frac{3}{4}}{1-{\left(\frac{3}{4}\right)}^2}\\ =\frac{\frac{3}{2}}{1-\frac{9}{16}}\\ \end{array}$

  $\begin{array}{c}=\frac{\frac{3}{2}}{\frac{7}{16}}\\ =\frac{24}{7}\end{array}$ .

Therefore the exact value of $\tan 2\theta$ is $\frac{24}{7}$ .

Substitute this value in the formula

  $\begin{array}{c}\cot 2\theta =\frac{1}{\tan 2\theta }\\ =\frac{1}{\frac{24}{7}}\\ =\frac{7}{24}\end{array}$ .

Therefore the exact value of $\cot 2\theta$ is $\frac{7}{24}$ .