Problem 31 - Precalculus : Building Concepts and Connections

Question

Find the exact value of each trigonometric function. $$\tan 60^{\circ}$$

Step-by-Step Solution

Answer

The exact value of \(\tan 60^{\circ}\) is \(\sqrt{3}\).

Step 1: Recall the properties of special triangles and trigonometric functions

Recall the trigonometric functions for the specific values of angles. For a 30-60-90 triangle, the sides are in the ratio \(1 : \sqrt{3} : 2\) where the side opposite the 30 degree angle is 1, the side opposite the 60 degree angle is \(\sqrt{3}\), and the side opposite the 90 degree angle (the hypotenuse) is 2. Among the six trigonometric functions, tangent or '\(\tan\)' is the ratio of the side opposite to the angle to the adjacent side in a right-angled triangle.

Step 2: Calculate the tangent of the given angle

At 60 degrees, in a 30-60-90 triangle, the length of the side opposite the angle is \(\sqrt{3}\) and the length of the adjacent side is 1. The \(\tan\) function is defined as the ratio of these two sides within a right-angled triangle: \(\tan = \frac{opposite}{adjacent} = \frac{\sqrt{3}}{1}\).

Key Concepts

Special Triangles

Special triangles, like the 30-60-90 triangle, are an essential part of understanding trigonometric functions. These triangles have uniquely defined angle measures, which translate into fixed side ratios. In a 30-60-90 triangle, the sides follow the distinct ratio of \(1 : \sqrt{3} : 2\). This means that:

• The side opposite the 30-degree angle has a length of 1.
• The side opposite the 60-degree angle has a length of \(\sqrt{3}\).
• The hypotenuse, opposite the 90-degree angle, measures 2.

The importance of special triangles lies in their ability to provide quick and exact values for trigonometric functions without complicated calculations. Understanding these triangles helps you easily determine values such as sine, cosine, and, in our specific case, tangent for particular angles like 30, 45, and 60 degrees.
Recognizing these patterns also aids in memorizing and understanding the properties of trigonometry better.

Tangent Function

The tangent function, often denoted as \(\tan\), is one of the basic trigonometric functions. In a right-angled triangle, the tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side.
Mathematically, it's expressed as:

• \(\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\)

In our example of a 30-60-90 triangle, calculating \(\tan 60^{\circ}\) involves understanding this ratio concept. Specifically, for 60 degrees in this triangle:

• The side opposite is \(\sqrt{3}\).
• The side adjacent is 1.
• Thus, \(\tan 60^{\circ} = \frac{\sqrt{3}}{1} = \sqrt{3}\).

This simple calculation shows the power of understanding the tangent function as it emphasizes the relationship between the sides of the triangle for specific angles.

Right-Angled Triangle

Right-angled triangles are a cornerstone in trigonometry. Defined by having one of their angles as a 90-degree angle, these triangles are instrumental in forming the basis of trigonometric identities and functions.
When analyzing right-angled triangles:

• The largest side is the hypotenuse, always opposite the right angle.
• The other two sides are known as the opposite and adjacent sides based on the angle of interest.

Understanding the relationships between these sides is essential for using trigonometric functions like sine, cosine, and tangent. For example, knowing that \(\tan 60^{\circ}\) equals \(\frac{\sqrt{3}}{1}\) in a right-angled triangle showcases how these side relationships allow for straightforward function evaluations.
Moreover, right-angled triangles are not just abstract concepts; they are applied in numerous real-life contexts, from engineering to physics, making them an invaluable tool for problem-solving.