Question

`y=cot^-1(-sqrt3)` Find the exact value of y?

Find the exact value of y. Do not use a calculator.

Answer 1

The answer is `y=(5pi)/6+pik` where `k` is any integer.

Explanation:

`y=cot^-1(-sqrt3)`

`cot(y)=cancel(cot)(cancel(cot^-1)(-sqrt3))`

`cot(y)=-sqrt3`

To make the problem a bit easier to visualize, I'll convert `cot` to `tan` by raising both sides of the equation to the power of `-1`.

`(cot(y))^-1=(-sqrt3)^-1`

`1/cot(y)=1/sqrt3`

`tan(y)=1/(-sqrt3)="opposite"/"adjacent"`

We can now construct a diagram:

When you divide all the side lengths in this triangle by `2`, you get a familiar looking triangle:

A triangle with the side length ratio `1/2:sqrt3/2:1` is a `30^@:60^@:90^@` triangle. That means that our reference angle is `30^@`.

Since the horizontal side length is negative, that means that the actual angle is the supplement of `30^@`, which is `150^@`.

The calculation would remain the same after any rotation of `180^@` degrees, so you can express the full answer as `150^@+180k^@` to represent the original answer and any `180^@` rotation.

In radians, this is `(5pi)/6+pik`.