Problem 192 - Precalculus

Question

For the following exercises, find the exact value of each expression. $$ \tan \frac{\pi}{3} $$

Step-by-Step Solution

Answer

The exact value of \( \tan \frac{\pi}{3} \) is \( \sqrt{3} \).

Step 1: Understand the Task

We need to find the exact value of the tangent function, \( \tan \) for the angle \( \frac{\pi}{3} \). This angle is a common angle in trigonometry, known as 60 degrees in the degree system.

Step 2: Recall the Tangent Value for Common Angles

Remember the trigonometric values for common angles. For \( \tan \theta \), the value for \( \theta = \frac{\pi}{3} \) is known to be \( \sqrt{3} \). These values come from the unit circle or from the 30-60-90 right triangle.

Step 3: Use Right Triangle Trigonometry

In a 30-60-90 triangle, the sides are in the ratio 1 : \( \sqrt{3} \) : 2. For \( \tan \frac{\pi}{3} \), \( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \). The "opposite" side for \( 60^{\circ} \) is \( \sqrt{3} \) and the "adjacent" side is 1, so \( \tan \frac{\pi}{3} = \frac{\sqrt{3}}{1} = \sqrt{3} \).

Step 4: Confirm the Result

Cross-verify with other sources if \( \tan \frac{\pi}{3} = \sqrt{3} \) by checking a unit circle or trigonometry table.

Key Concepts

Tangent Function

The tangent function is a fundamental trigonometric function. It is often abbreviated as \( \tan \). This function relates to the ratio of two specific sides of a right triangle.
In mathematical terms, for a given angle \( \theta \), \( \tan \theta \) is defined as the ratio of the length of the opposite side to the length of the adjacent side.

• Opposite: The side opposite to the angle \( \theta \).
• Adjacent: The side next to the angle \( \theta \).

The tangent function is particularly useful because it provides a way to calculate the slope or steepness of an angle. In this case of \( \tan \frac{\pi}{3} \), since the opposite and adjacent sides of the 60° angle are known, we find \( \tan \frac{\pi}{3} = \sqrt{3} \). Understanding the tangent function and its relationship with right triangles is a crucial step in mastering trigonometry.

Unit Circle

The unit circle is a powerful tool in trigonometry, representing angles as coordinates on a circle with radius 1. It simplifies understanding of trigonometric functions.
The angle \( \frac{\pi}{3} \) or 60 degrees on the unit circle corresponds to a specific point with coordinates \( (\frac{1}{2}, \frac{\sqrt{3}}{2}) \). This connection between angles and coordinate points helps us visualize and calculate trigonometric values.

• The x-coordinate relates to the cosine function.
• The y-coordinate relates to the sine function.
• The tangent can be found as \( \tan \theta = \frac{\text{sine}}{\text{cosine}} \).

For \( \theta = \frac{\pi}{3} \), this becomes \( \tan \theta = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \sqrt{3} \). Thus, the unit circle not only helps in memorizing these values but also in understanding their derivation.

30-60-90 Triangle

A 30-60-90 triangle is a special kind of right triangle, which makes it highly useful in trigonometry. It is known for its predictable side ratios, which are always \( 1 : \sqrt{3} : 2 \).
This triangle is derived by cutting an equilateral triangle in half. That's why these angles are consistently 30°, 60°, and 90°.

• In a 30-60-90 triangle, the shortest side (opposite the 30° angle) is \( 1 \).
• The longer leg (opposite the 60° angle) is \( \sqrt{3} \).
• The hypotenuse is \( 2 \).

For the task of finding \( \tan \frac{\pi}{3} \), or \( \tan 60° \), we use these side ratios. Since tangent is the opposite over adjacent, for 60°, it is \( \sqrt{3} / 1 = \sqrt{3} \). This confirms the result obtained from previous methods, like using the unit circle.