Question

Answer question number 30. The question is in the image.

Answer 1

#30

We need to find the value of

`cot((-8\pi)/(3))`

1. We will add 2pi to it until change it from negative to positive, then look for its quadrant

`\begin{gathered} (-8\pi)/(3)+2\pi=(-8\pi)/(3)+(6\pi)/(3)=(-2\pi)/(3) \\ \\ (-2\pi)/(3)+2\pi=(-2\pi)/(3)+(6\pi)/(3)=(4\pi)/(3) \end{gathered}`

The angle 4pi/3 lies on the 3rd quadrant

Since the angle in the 3rd quadrant has the form

`\pi+\theta`

Where theta is an acute angle, then

`\begin{gathered} \pi+\theta=(4)/(3)\pi \\ \theta=(4\pi)/(3)-\pi \\ \theta=(4\pi)/(3)-(3\pi)/(3) \\ \theta=(\pi)/(3) \end{gathered}`

Then we will find the value of tan(pi/3), then reciprocal it to find cot

`tan(\pi)/(3)=√(3)`

Reciprocal it

`cot((\pi)/(3))=(1)/(√(3))*(√(3))/(√(3))=(√(3))/(3)`

In the 3rd quadrant tan and cot are positive values, then

The value of

`\begin{gathered} cot(\pi+(\pi)/(3))=(√(3))/(3) \\ \\ cot((4\pi)/(3))=(√(3))/(3) \\ \\ cot((-8\pi)/(3))=(√(3))/(3) \end{gathered}`

The answer is

`(√(3))/(3)`