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:

`(7\pi)/(6)radians=(7\pi)/(6)radians\cdot\frac{180\text{ \degree}}{\pi\text{ radians}}=210\text{ \degree}`

From the table:

`\sin ((7\pi)/(6))=\sin (210^o)=-(1)/(2)`

8π/3 can be expressed as follows:

`(8)/(3)\pi=2\pi+(2)/(3)\pi`

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

`\tan ((8)/(3)\pi)`

is the same as evaluating:

`\tan ((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:

`(2\pi)/(3)radians=(2\pi)/(3)radians\cdot\frac{180\text{ \degree}}{\pi\text{ radians}}=120\text{ \degree}`

From the table:

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

Substituting these values into the original expression:

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