# cot of pi/3

#### Understand the Problem

The question is asking for the value of the cotangent of the angle pi/3. The cotangent function is the reciprocal of the tangent function, and this will involve evaluating the tangent of pi/3 and then taking the reciprocal of that value.

$\frac{\sqrt{3}}{3}$

The value of the cotangent of the angle $\frac{\pi}{3}$ is $\frac{\sqrt{3}}{3}$.

#### Steps to Solve

1. Identify the Angle We are tasked with finding the cotangent of the angle $\frac{\pi}{3}$.

2. Evaluate the Tangent Function The cotangent function is the reciprocal of the tangent function. We first need to find $\tan\left(\frac{\pi}{3}\right)$. The tangent of $\frac{\pi}{3}$ can be evaluated using the sine and cosine:

$$\tan\left(\frac{\pi}{3}\right) = \frac{\sin\left(\frac{\pi}{3}\right)}{\cos\left(\frac{\pi}{3}\right)}$$

1. Find Sine and Cosine Values The sine and cosine values for the angle $\frac{\pi}{3}$ are:

$$\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}$$

$$\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}$$

Using these values, we calculate:

$$\tan\left(\frac{\pi}{3}\right) = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \sqrt{3}$$

1. Calculate Cotangent Now we calculate cotangent by taking the reciprocal of the tangent:

$$\cot\left(\frac{\pi}{3}\right) = \frac{1}{\tan\left(\frac{\pi}{3}\right)} = \frac{1}{\sqrt{3}}$$

1. Rationalize the Denominator To express this in a more standard form, we multiply the numerator and denominator by $\sqrt{3}$:

$$\cot\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{3}$$

The value of the cotangent of the angle $\frac{\pi}{3}$ is $\frac{\sqrt{3}}{3}$.