Question

What is cosecant and cotangent of pi/3?

Answer 1

`csc(\pi/3) = 2/\sqrt{3}` and `cot(\pi/3) = 1/sqrt{3}`

Explanation:

`\pi/3` is `60^\circ` so this is trig's biggest cliche, the 30,60,90 right triangle, half an equilateral triangle. The hypotenuse is 1, the short side adjacent to the `30^\circ` angle is `1/2` and by the Pythagorean Thoerem the long side is `\sqrt{1^2 - (1/2)^2} = \sqrt{3}/2`

`\cos(\pi/3)=cos(60^circ)= frac{text{adjacent}}{text{hypotenuse}} = 1 / 2`

`\sin(\pi/3) = \sin(60^\circ) = frac{text{opposite}}{text{hypotenuse}} = sqrt{3} /2`

`\csc(\pi/3) = 1/\sin(\pi/3) = 2/\sqrt{3} = 2 /3 sqrt{3}`

`cot(\pi/3) = frac{cos(pi/3)}{sin(pi/3)} = frac{1/2}{\sqrt{3}/2} = 1/\sqrt{3}`