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
- 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:
$$ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} $$
- Find Sine and Cosine Values
Using known values from the unit circle, we know:
$$ \sin(30^\circ) = \frac{1}{2} $$ $$ \cos(30^\circ) = \frac{\sqrt{3}}{2} $$
- 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:
$$ \tan(30^\circ) = \frac{1}{\sqrt{3}} $$
- 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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