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

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Answer 1 · 2018-03-16T15:42:45

$arctan (-sqrt3) = 120° or 300°$

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:
$360°-60° = 300°$

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