Question

How do you find the exact value of `Arctan -sqrt3`?

Answer 1

`arctan(-sqrt3)=-pi/3`

Explanation:

First, recognize that the domain of the function `arctan(x)` is `-pi/2 < x < pi/2`.

Now, a little bit about what `arctan(x)` means:

`arctan(x)` and `tan(x)` are inverse functions.

This means that `mathbf(arctan(tan(x))=x` and `mathbf(tan(arctan(x))=x`.

We can see tangent and arctangent as "undoing" one another. Another way we can see `arctan(x)` is as `tan(x)` in reverse.

We know that `tan(pi/4)=1`. This means that `arctan(1)=pi/4`.

We see that:

• in `tan(x)`, `x` is an angle
• in `arctan(x)`, `x` is the value of the tangent function

So, back to the original question:

What is the value of `arctan(-sqrt3)`?

This is essentially asking, the tangent of what angle gives `mathbf(-sqrt3)` ?

Since `tan(pi/3)=sqrt3`, we know that `tan(-pi/3)=-sqrt3`.

We can reverse this with the `arctan` function to see that

`arctan(-sqrt3)=-pi/3`