How do you evaluate cot (- pi/6)?

1 Answer
sente
Nov 13, 2015

cot(-pi/6) = -sqrt(3)

Explanation:

We will use the following:
(1) cot(x) = cos(x)/sin(x)
(2) sin(-x) = -sin(x)
(3) cos(-x) = cos(x)
(4) cos(pi/6) = sqrt(3)/2
(5) sin(pi/6) = 1/2
Verification of these facts is a good exercise, and may be done using basic definitions together with the unit circle for 1-3.
For 4 and 5, as a hint, bisect an equilateral triangle with side length 1 into two equal right triangles. What are the angles of the right triangles?

With this, we have
cot(-pi/6) =cos(-pi/6)/sin(-pi/6) (by 1)
=>cot(-pi/6) = cos(pi/6)/(-sin(pi/6)) (by 2 and 3)
=>cot(-pi/6) = -(sqrt(3)/2)/(1/2) = -sqrt(3) (by 4 and 5)

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