How do you evaluate arctan(-sqrt3)arctan(3) without a calculator?

1 Answer
Mar 16, 2018

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°