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.

Answer

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

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

More Information

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.

Tips

  • A common mistake is to confuse the values of sine and cosine for 30 degrees and 45 degrees. Always ensure that you are using the correct angle values.
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