Chapter T, Problem 4DDT

(a)

To determine

To find: The exact value of $\tan \left(\frac{\pi }{3}\right)$.

Answer to Problem 4DDT

The exact value of $\tan \left(\frac{\pi }{3}\right)$ is $\sqrt{3}$.

Explanation of Solution

Result used:

The trigonometry result is $\tan x=\frac{\sin x}{\cos x}$.

Calculation:

The value of $\frac{\pi }{3}$ is $60°$.

Substitute $\frac{\pi }{3}=60°$ in the above mentioned result.

$\begin{array}{c}\tan 60°=\frac{\sin 60°}{\cos 60°}\\ =\frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}}\\ =\frac{\sqrt{3}}{2}\times 2\\ =\sqrt{3}\end{array}$

Thus, the exact value of $\tan \left(\frac{\pi }{3}\right)$ is $\sqrt{3}$.

(b)

To determine

To find: The exact value of $\sin \left(\frac{7\pi }{6}\right)$.

Answer to Problem 4DDT

The exact value of $\sin \left(\frac{7\pi }{6}\right)$ is $-\frac{1}{2}$.

Explanation of Solution

Result used:

The trigonometry result is $\sin \left(\pi +x\right)=-\sin x$.

Calculation:

Use the above mentioned result and solve the exact value of $\sin \left(\frac{7\pi }{6}\right)$ as follows,

$\begin{array}{c}\sin \left(\frac{7\pi }{6}\right)=\sin \left(\pi +\frac{\pi }{6}\right)\\ =-\sin \left(\frac{\pi }{6}\right)\\ =-\frac{1}{2}\end{array}$

Thus, the exact value of $\sin \left(\frac{7\pi }{6}\right)$ is $-\frac{1}{2}$.

(c)

To determine

To find: The exact value of $\sec \left(\frac{5\pi }{3}\right)$.

Answer to Problem 4DDT

The exact value of $\sec \left(\frac{5\pi }{3}\right)$ is $2$.

Explanation of Solution

Result used:

The trigonometry result is $\sec \left(2\pi -x\right)=\sec x$.

Calculation:

Use the above mentioned result and solve the exact value of $\sec \left(\frac{5\pi }{3}\right)$ as follows,

$\begin{array}{c}\sec \left(\frac{5\pi }{3}\right)=\sec \left(2\pi -\frac{\pi }{3}\right)\\ =\sec \left(\frac{\pi }{3}\right)\\ =\frac{1}{\cos \left(\frac{\pi }{3}\right)}\\ =\frac{1}{\frac{1}{2}}\end{array}$

$=2$

Thus, the exact value of $\sec \left(\frac{5\pi }{3}\right)$ is 2.