Chapter 5.5, Problem 10E

(a)

To determine

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

Answer to Problem 10E

The exact value of the trigonometric function $\sin \theta$ is $\frac{12}{13}$ .

Explanation of Solution

Given:

The figure is given as:

  

Concept Used:

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

Calculation:

As $\text{opp}=12$ , $\text{adj}=5$ . Now use the Pythagoras theorem

  $\begin{array}{c}\text{hyp}=\sqrt{{\left(\text{adj}\right)}^2+{\left(\text{opp}\right)}^2}\\ =\sqrt{{\left(5\right)}^2+{\left(12\right)}^2}\\ =\sqrt{25+144}\\ =\sqrt{169}\\ =13\end{array}$ .

Therefore the value of $\text{hyp}=13$ .

Now substitute these values in the formula

  $\begin{array}{c}\sin \theta =\frac{\text{opp}}{\text{hyp}}\\ =\frac{12}{13}\end{array}$ .

Therefore, the exact value of $\sin \theta$ is $\frac{12}{13}$ .

(b)

To determine

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

Answer to Problem 10E

The exact value of the trigonometric function $\cos \theta$ is $\frac{5}{13}$ .

Explanation of Solution

Given:

The figure is given as:

  

Concept Used:

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

Calculation:

As $\text{opp}=12$ , $\text{adj}=5$ . Now use the Pythagoras theorem

  $\begin{array}{c}\text{hyp}=\sqrt{{\left(\text{adj}\right)}^2+{\left(\text{opp}\right)}^2}\\ =\sqrt{{\left(5\right)}^2+{\left(12\right)}^2}\\ =\sqrt{25+144}\\ =\sqrt{169}\\ =13\end{array}$ .

Therefore the value of $\text{hyp}=13$ .

Now substitute these values in the formula

  $\begin{array}{c}\cos \theta =\frac{\text{adj}}{\text{hyp}}\\ =\frac{5}{13}\end{array}$ .

Therefore, the exact value of $\cos \theta$ is $\frac{5}{13}$ .

(c)

To determine

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

Answer to Problem 10E

The exact value of the trigonometric function $\tan \theta$ is $\frac{12}{5}$ .

Explanation of Solution

Given:

The figure is given as:

  

Concept Used:

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

Calculation:

As $\text{opp}=12$ , $\text{adj}=5$ . Now use the Pythagoras theorem

  $\begin{array}{c}\text{hyp}=\sqrt{{\left(\text{adj}\right)}^2+{\left(\text{opp}\right)}^2}\\ =\sqrt{{\left(5\right)}^2+{\left(12\right)}^2}\\ =\sqrt{25+144}\\ =\sqrt{169}\\ =13\end{array}$ .

Therefore the value of $\text{hyp}=13$ .

Now substitute these values in the formula

  $\begin{array}{c}\tan \theta =\frac{\text{opp}}{\text{adj}}\\ =\frac{12}{5}\end{array}$ .

Therefore, the exact value of $\tan \theta$ is $\frac{12}{5}$ .

(d)

To determine

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

Answer to Problem 10E

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

Explanation of Solution

Given:

The figure is given as:

  

Concept Used:

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

Calculation:

As $\text{opp}=12$ , $\text{adj}=5$ . Now use the Pythagoras theorem

  $\begin{array}{c}\text{hyp}=\sqrt{{\left(\text{adj}\right)}^2+{\left(\text{opp}\right)}^2}\\ =\sqrt{{\left(5\right)}^2+{\left(12\right)}^2}\\ =\sqrt{25+144}\\ =\sqrt{169}\\ =13\end{array}$ .

Therefore the value of $\text{hyp}=13$ .

Now substitute these values in the formula

  $\begin{array}{c}\sin \theta =\frac{\text{opp}}{\text{hyp}}\\ =\frac{12}{13}\end{array}$ .

Therefore, the exact value of $\sin \theta$ is $\frac{12}{13}$ .

Now substitute the value in the formula

  $\begin{array}{c}\cos 2\theta =1-2{\sin }^2\theta \\ =1-2{\left(\frac{12}{13}\right)}^2\\ =1-2\left(\frac{144}{169}\right)\\ =1-\frac{288}{169}\\ =-\frac{119}{169}\end{array}$ .

Therefore the exact value of $\cos 2\theta$ is $-\frac{119}{169}$ .

(e)

To determine

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

Answer to Problem 10E

The exact value of the trigonometric function $\tan 2\theta$ is $-\frac{120}{119}$ .

Explanation of Solution

Given:

The figure is given as:

  

Concept Used:

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

Calculation:

As $\text{opp}=12$ , $\text{adj}=5$ . Now use the Pythagoras theorem

  $\begin{array}{c}\text{hyp}=\sqrt{{\left(\text{adj}\right)}^2+{\left(\text{opp}\right)}^2}\\ =\sqrt{{\left(5\right)}^2+{\left(12\right)}^2}\\ =\sqrt{25+144}\\ =\sqrt{169}\\ =13\end{array}$ .

Therefore the value of $\text{hyp}=13$ .

Now substitute these values in the formula

  $\begin{array}{c}\tan \theta =\frac{\text{opp}}{\text{adj}}\\ =\frac{12}{5}\end{array}$ .

Therefore, the exact value of $\tan \theta$ is $\frac{12}{5}$ .

Substitute the value in the formula

  $\begin{array}{c}\tan 2\theta =\frac{2\tan \theta }{1-{\tan }^2\theta }\\ =\frac{2\times \frac{12}{5}}{1-{\left(\frac{12}{5}\right)}^2}\\ =\frac{\frac{24}{5}}{1-\frac{144}{25}}\\ \end{array}$

  $\begin{array}{c}=\frac{\frac{24}{5}}{\frac{-119}{25}}\\ =-\frac{120}{119}\end{array}$ .

Therefore the exact value of $\tan 2\theta$ is $-\frac{120}{119}$ .

(f)

To determine

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

Answer to Problem 10E

The exact value of the trigonometric function $\cot 2\theta$ is $-\frac{119}{120}$ .

Explanation of Solution

Given:

The figure is given as:

  

Concept Used:

Use the formula $\cot 2\theta =\frac{1}{\tan 2\theta }$ .

Calculation:

As $\text{opp}=12$ , $\text{adj}=5$ . Now use the Pythagoras theorem

  $\begin{array}{c}\text{hyp}=\sqrt{{\left(\text{adj}\right)}^2+{\left(\text{opp}\right)}^2}\\ =\sqrt{{\left(5\right)}^2+{\left(12\right)}^2}\\ =\sqrt{25+144}\\ =\sqrt{169}\\ =13\end{array}$ .

Therefore the value of $\text{hyp}=13$ .

Now substitute these values in the formula

  $\begin{array}{c}\tan \theta =\frac{\text{opp}}{\text{adj}}\\ =\frac{12}{5}\end{array}$ .

Therefore, the exact value of $\tan \theta$ is $\frac{12}{5}$ .

Substitute the value in the formula

  $\begin{array}{c}\tan 2\theta =\frac{2\tan \theta }{1-{\tan }^2\theta }\\ =\frac{2\times \frac{12}{5}}{1-{\left(\frac{12}{5}\right)}^2}\\ =\frac{\frac{24}{5}}{1-\frac{144}{25}}\\ \end{array}$

  $\begin{array}{c}=\frac{\frac{24}{5}}{\frac{-119}{25}}\\ =-\frac{120}{119}\end{array}$ .

Therefore the exact value of $\tan 2\theta$ is $-\frac{120}{119}$ .

Substitute this value in the formula

  $\begin{array}{l}\cot 2\theta =\frac{1}{\tan 2\theta }\\ =\frac{1}{-\frac{120}{119}}\\ =-\frac{119}{120}\end{array}$ .

Therefore the exact value of $\cot 2\theta$ is $-\frac{119}{120}$ .

(g)

To determine

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

Answer to Problem 10E

The exact value of the trigonometric function $\sec 2\theta$ is $-\frac{169}{119}$ .

Explanation of Solution

Given:

The figure is given as:

  

Concept Used:

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

Calculation:

As $\text{opp}=12$ , $\text{adj}=5$ . Now use the Pythagoras theorem

  $\begin{array}{c}\text{hyp}=\sqrt{{\left(\text{adj}\right)}^2+{\left(\text{opp}\right)}^2}\\ =\sqrt{{\left(5\right)}^2+{\left(12\right)}^2}\\ =\sqrt{25+144}\\ =\sqrt{169}\\ =13\end{array}$ .

Therefore the value of $\text{hyp}=13$ .

Now substitute these values in the formula

  $\begin{array}{c}\sin \theta =\frac{\text{opp}}{\text{hyp}}\\ =\frac{12}{13}\end{array}$ .

Therefore, the exact value of $\sin \theta$ is $\frac{12}{13}$ .

Now substitute the value in the formula

  $\begin{array}{c}\cos 2\theta =1-2{\sin }^2\theta \\ =1-2{\left(\frac{12}{13}\right)}^2\\ =1-2\left(\frac{144}{169}\right)\\ =1-\frac{288}{169}\\ =-\frac{119}{169}\end{array}$ .

Therefore the exact value of $\cos 2\theta$ is $-\frac{119}{169}$ .

Substitute this value in the formula

  $\begin{array}{c}\sec 2\theta =\frac{1}{\cos 2\theta }\\ =\frac{1}{-\frac{119}{169}}\\ =-\frac{169}{119}\end{array}$ .

Therefore the exact value of $\sec 2\theta$ is $-\frac{169}{119}$ .

(h)

To determine

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

Answer to Problem 10E

The exact value of the trigonometric function $\csc 2\theta$ is $\frac{169}{120}$ .

Explanation of Solution

Given:

The figure is given as:

  

Concept Used:

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

Calculation:

As $\text{opp}=12$ , $\text{adj}=5$ . Now use the Pythagoras theorem

  $\begin{array}{c}\text{hyp}=\sqrt{{\left(\text{adj}\right)}^2+{\left(\text{opp}\right)}^2}\\ =\sqrt{{\left(5\right)}^2+{\left(12\right)}^2}\\ =\sqrt{25+144}\\ =\sqrt{169}\\ =13\end{array}$ .

Therefore the value of $\text{hyp}=13$ .

Now substitute these values in the formula

  $\begin{array}{c}\sin \theta =\frac{\text{opp}}{\text{hyp}}\\ =\frac{12}{13}\end{array}$ .

Therefore, the exact value of $\sin \theta$ is $\frac{12}{13}$ .

Now substitute the values in the formula

  $\begin{array}{c}\cos \theta =\frac{\text{adj}}{\text{hyp}}\\ =\frac{5}{13}\end{array}$ .

Therefore the exact value of $\cos \theta$ is $\frac{5}{13}$ .

Now substitute both the values in the formula

  $\begin{array}{c}\sin 2\theta =2\sin \theta \cos \theta \\ =2\times \frac{5}{13}\times \frac{12}{13}\\ =\frac{120}{169}\end{array}$ .

Therefore the exact value of $\sin 2\theta$ is $\frac{120}{169}$ .

Substitute this value in the formula

  $\begin{array}{c}\csc 2\theta =\frac{1}{\sin 2\theta }\\ =\frac{1}{\frac{120}{169}}\\ =\frac{169}{120}\end{array}$ .

Therefore the exact value of $\csc 2\theta$ is $\frac{169}{120}$ .