Question

How do you find the value of Cot(2pi/3)?

Answer 1

By breaking it down into its most basic form, `cos(theta)/(sin(theta))`. The answer, by the way, is `-sqrt3/3`.

Explanation:

So we know two trigonometric functions, our old friends sine and cosine. Everything else is a derivation of these. Tangent, for instance, is sine over cosine, or `sin(theta)/(cos(theta))`.

The fundamental functions have reciprocal functions , which are their inverse . The reciprocate of sine is cosecant, the reciprocal of cosine is secant, and the reciprocal of tangent is cotangent.

If tangent is `sin(theta)/(cos(theta))`, then cotangent is one over that, or `cos(theta)/(sin(theta))` .

Now we must remember a little handy device called the unit circle. The unit circle is a circle of radius one, and in trigonometry it is contained in the Cartesian coordinate plane. The x-axis is cosine and the y-axis is sine . It looks like this:

This device is very useful for a variety of purposes. You can see we marked some notable angles in the circle, and they are associated with their respective sine and cosine values. These angles can be expressed in degrees or in radians.

To convert from one unit to the other, remember:

`pi rightarrow 180˚`

You can easily see where `(2pi)/3` is: it's in the second quadrant, meaning that its sine is positive and its cosine is negative. In degrees, it is equal to 120˚ - being the supplementary angle of 60˚ (`pi/3`), it has the same sine value as 60˚ and the opposite cosine value.

That means `sin((2pi)/3) = (sqrt3)/2` and `cos((2pi)/3) = -1/2`.

We want to know its cotangent, so:

`cot((2pi)/3) = cos((2pi)/3)/(sin((2pi)/3)) =`

`= (-1/2)/(sqrt3/2) =`

`= -1/cancel2*cancel2/sqrt3 =`

`= -1/sqrt3`

That's not a very pretty number, but we can rationalise the denominator so that it no longer contains a square root:

`= -1/sqrt3 * sqrt3/sqrt3 =`

`= -sqrt3/3`

Hope this helped!