Question

How do you evaluate `arctan(-sqrt3)` without a calculator?

Answer 1

`arctan (-sqrt3) = 120° or 300°`

Explanation:

One of the special triangles in Geometry and Trig is the right-angled triangle with angles of `30°, 60° and 90°`

The sides of this triangle are in the ratio :`1" ":" "2" ":" "sqrt3`

For the angle of `60°` the opposite side is `sqrt3`

`sqrt3= sqrt3/1` indicating the lengths of the opposite and the adjacent sides.

The basic angle is therefore `60°`

`arctan( sqrt3/1) =60°`

However, in this case we have `-sqrt3`

Tan is negative in the 2nd and 4th quadrants.

In the 2nd quadrant, the angles are `180°-theta`
which gives `180°60° = 120°`

In the 4th quadrant, the angles are `360-theta` which gives:
`360°-60° = 300°`