Question

Find the exact value of each of the following How do I finish c and d

Answer 1

c) First, we need to convert 7π/6 radians to degrees. π radians are equivalent to 180°, then:

$\frac{7\pi}{6}radians=\frac{7\pi}{6}radians\cdot\frac{180\text{ \degree}}{\pi\text{ radians}}=210\text{ \degree}$

From the table:

$\sin (\frac{7\pi}{6})=\sin (210^o)=-\frac{1}{2}$

8π/3 can be expressed as follows:

$\frac{8}{3}\pi=2\pi+\frac{2}{3}\pi$

The function tan(x) is periodic, with a period of π. This means that evaluating:

$\tan (\frac{8}{3}\pi)$

is the same as evaluating:

$\tan (\frac{2}{3}\pi)$

In this case, x (the input in the function) is translated 2π units to the left. From the periodicity of the function, the values are the same.

2π/3 radians is converted to degrees as follows:

$\frac{2\pi}{3}radians=\frac{2\pi}{3}radians\cdot\frac{180\text{ \degree}}{\pi\text{ radians}}=120\text{ \degree}$

From the table:

$\tan (\frac{2}{3}\pi)=\tan (120^o)=-\sqrt[]{3}$

Substituting these values into the original expression:

$\begin{gathered} \sin (\frac{7\pi}{6})\cdot\tan (\frac{8}{3}\pi)= \\ =\sin (\frac{7\pi}{6})\cdot\tan (\frac{2}{3}\pi)= \\ =(-\frac{1}{2})\cdot(-\sqrt[]{3})= \\ =\frac{\sqrt[]{3}}{2} \end{gathered}$