# What is the exact value of tan 60 degrees?

#### Understand the Problem

The question is asking for the exact value of the tangent of 60 degrees, which is a trigonometric function commonly used in mathematics. To solve this, we recall that tan(60°) is a well-known value based on the properties of a 30-60-90 triangle.

$\sqrt{3}$

The exact value of $\tan(60°)$ is $\sqrt{3}$.

#### Steps to Solve

1. Recall the properties of a 30-60-90 triangle

A 30-60-90 triangle has specific side lengths based on the angles:

• The length of the side opposite the 30-degree angle is $x$.
• The length of the side opposite the 60-degree angle is $x\sqrt{3}$.
• The length of the hypotenuse is $2x$.
1. Identify the tangent function

The tangent of an angle in a right triangle is defined as the ratio of the length of the opposite side to the length of the adjacent side. For 60 degrees, we have: $$\tan(60°) = \frac{\text{opposite}}{\text{adjacent}}$$

1. Substitute the appropriate side lengths

In our 30-60-90 triangle, the opposite side to 60 degrees is $x\sqrt{3}$, and the adjacent side (which is opposite the 30 degrees) is $x$. Thus: $$\tan(60°) = \frac{x\sqrt{3}}{x}$$

1. Simplify the equation

Cancelling out the $x$ in the numerator and denominator, we arrive at: $$\tan(60°) = \sqrt{3}$$

The exact value of $\tan(60°)$ is $\sqrt{3}$.

The tangent of 60 degrees, $\sqrt{3}$, is commonly found in trigonometry and often arises in problems involving 30-60-90 right triangles. This value is frequently used in physics and engineering.
• Confusing the tangent of 60 degrees with the tangent of 30 degrees. Remember: $\tan(30°) = \frac{1}{\sqrt{3}}$ while $\tan(60°) = \sqrt{3}$.