Chapter 7, Problem 3CRT

(a)

To determine

To find:The exact value of $\cos \frac{7\pi }{6}$ .

Answer to Problem 3CRT

The exact value of $\cos \frac{7\pi }{6}$ is $-\frac{\sqrt{3}}{2}$ .

Explanation of Solution

Given information:

The trigonometric value is $\cos \frac{7\pi }{6}$ .

Formula used:

Use the trigonometric identity $\cos \left(\pi +\theta \right)=-\cos \theta$ .

Calculation:

Substitute $\frac{\pi }{6}$ for $\theta$ in the identity.

  $\begin{array}{c}\cos \left(\pi +\theta \right)=-\cos \theta \\ \cos \left(\pi +\frac{\pi }{6}\right)=-\cos \frac{\pi }{6}\\ \cos \left(\frac{7\pi }{6}\right)=-\cos \frac{\pi }{6}\end{array}$

Substitute $\frac{\sqrt{3}}{2}$ for $\cos \frac{\pi }{6}$ in the above equation.

  $\begin{array}{c}\cos \left(\frac{7\pi }{6}\right)=-\cos \frac{\pi }{6}\\ \cos \left(\frac{7\pi }{6}\right)=-\frac{\sqrt{3}}{2}\end{array}$

Therefore, the exact value of $\cos \frac{7\pi }{6}$ is $-\frac{\sqrt{3}}{2}$ .

(b)

To determine

To find:The exact value of $\tan 135°$ .

Answer to Problem 3CRT

The exact value of $\tan 135°$ is $-1$ .

Explanation of Solution

Given information:

The trigonometric value is $\tan 135°$ .

Formula used:

Use the trigonometric identity $\tan \left(180°-\theta \right)=-\tan \theta$ , for any acute angle $\theta$ .

Calculation:

Substitute $45°$ for $\theta$ in the identity.

  $\begin{array}{c}\tan \left(180°-\theta \right)=-\tan \theta \\ \tan \left(180°-45°\right)=-\tan 45°\\ \tan \left(135°\right)=-\tan 45°\end{array}$

Substitute $1$ for $\tan 45°$ in the above equation.

  $\begin{array}{c}\tan \left(135°\right)=-\tan 45°\\ \tan \left(135°\right)=-1\end{array}$

Therefore, the exact value of $\tan 135°$ is $-1$ .

(c)

To determine

To find:The exact value of $\csc 240°$ .

Answer to Problem 3CRT

The exact value of $\csc 240°$ is $-\frac{2\sqrt{3}}{3}$ .

Explanation of Solution

Given information:

The trigonometric value is $\csc 240°$ .

Formula used:

Use the trigonometric identity $\csc \left(180°+\theta \right)=-\csc \theta$ ,for any acute angle $\theta$ .

Calculation:

Substitute $60°$ for $\theta$ in the identity.

  $\begin{array}{c}\csc \left(180°+\theta \right)=-\csc \theta \\ \csc \left(180°+60°\right)=-\csc 60°\\ \csc \left(240°\right)=-\csc 60°\end{array}$

Substitute $\frac{2\sqrt{3}}{3}$ for $\csc 60°$ in the above equation.

  $\begin{array}{c}\csc \left(240°\right)=-\csc 60°\\ \csc \left(240°\right)=-\frac{2\sqrt{3}}{3}\end{array}$

Therefore, The exact value of $\csc 240°$ is $-\frac{2\sqrt{3}}{3}$ .

(d)

To determine

To find:The exact value of $\sin \left(-\frac{9\pi }{2}\right)$ .

Answer to Problem 3CRT

The exact value of $\sin \left(-\frac{9\pi }{2}\right)$ is $-1$ .

Explanation of Solution

Given information:

The trigonometric value is $\csc 240°$ .

Formula used:

Use the trigonometric identity $\sin \left(-\theta \right)=-\sin \theta$ .

Use another trigonometric identity $\sin \left(4\pi +\theta \right)=\sin \theta$ , for any acute angle $\theta$ .

Calculation:

Substitute $\frac{9\pi }{2}$ for $\theta$ in the first identity.

  $\begin{array}{c}\sin \left(-\theta \right)=-\sin \theta \\ \sin \left(-\frac{9\pi }{2}\right)=-\sin \frac{9\pi }{2}\end{array}$

Now substitute $\frac{\pi }{2}$ for $\theta$ the second identity.

  $\begin{array}{c}\sin \left(4\pi +\theta \right)=\sin \theta \\ \sin \left(4\pi +\frac{\pi }{2}\right)=\sin \frac{\pi }{2}\\ \sin \left(\frac{9\pi }{2}\right)=\sin \frac{\pi }{2}\end{array}$

Substitute $1$ for $\sin \frac{\pi }{2}$ in the above equation.

  $\begin{array}{c}\sin \left(\frac{9\pi }{2}\right)=\sin \frac{\pi }{2}\\ \sin \left(\frac{9\pi }{2}\right)=1\end{array}$

Substitute $1$ for $\sin \left(\frac{9\pi }{2}\right)$ in the equation.

  $\begin{array}{l}\sin \left(-\frac{9\pi }{2}\right)=-\sin \frac{9\pi }{2}\\ \sin \left(-\frac{9\pi }{2}\right)=-1\end{array}$

Therefore, the exact value of $\sin \left(-\frac{9\pi }{2}\right)$ is $-1$ .