Why is inverse tangent of -1 equal to pi/4?

Answer 1

It isn't. tan(pi/4) = 1, not -1. The range of arctan is −π/2 to π/2, so while there are infinitely many angles with tangent equal to -1, arctan(-1) = -π/4, the only value in the range of arctan that has tangent equal to -1.

Answer 2

The definition of the inverse tangent is as follows:

y = arctan(x) if and only if x = tan(y) for -pi/2 < y < pi/2

If we wanted to calculate arctan(-1), then it would be the same as calculating

tan(y) = -1

Tan is equal to -1 in the 2nd and 4th quadrants. Since -pi/2 to pi/2 encompasses the 1st and 4th quadrants, the solution must lie in the 4th quadrant AND be less than 0 (since, in the 4th quadrant, -pi/2 < y < 0).

On the restriction 0 < y < 2pi, the answer would be 3pi/4 and 7pi/4. That means we have to convert 7pi/4 to be in the 4th quadrant (done by subtracting 2pi, or 8pi/4). 7pi/4 - 8pi/4 is equal to -pi/4.

This means y = -pi/4.

Answer 3

It isn't. arctan (-1) = -π/4, not π/4.