# What is the exact value of tan 30 degrees?

#### Understand the Problem

The question is asking for the exact value of the tangent of 30 degrees, which is a trigonometric function value that can be derived from known angles in trigonometry.

The exact value of $\tan(30^\circ)$ is $\frac{\sqrt{3}}{3}$.

The exact value of $\tan(30^\circ)$ is $\frac{\sqrt{3}}{3}$.

#### Steps to Solve

1. Identify the Relationship with Trigonometric Functions

To find the tangent of 30 degrees, we can use the relationship between tangent and sine/cosine. The tangent function is defined as:

$$\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$$

1. Find Sine and Cosine Values

Using known values from the unit circle, we know:

$$\sin(30^\circ) = \frac{1}{2}$$ $$\cos(30^\circ) = \frac{\sqrt{3}}{2}$$

1. Calculate the Tangent Value

Now we can substitute these values into the formula:

$$\tan(30^\circ) = \frac{\sin(30^\circ)}{\cos(30^\circ)} = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}}$$

This simplifies to:

$$\tan(30^\circ) = \frac{1}{\sqrt{3}}$$

1. Rationalize the Denominator

To present the final answer in a more standard form, we can rationalize the denominator:

$$\tan(30^\circ) = \frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{3}}{3}$$

The exact value of $\tan(30^\circ)$ is $\frac{\sqrt{3}}{3}$.

The tangent function is often used in right triangle problems. The value of $\tan(30^\circ) = \frac{\sqrt{3}}{3}$ is an important reference that is derived from the properties of a 30-60-90 triangle, where the lengths of the sides have a specific ratio.