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Chapter 6: Problem 28

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

Short Answer

Expert verified
\( \tan 60^{\text{circ}} = \sqrt{3} \)

Step by step solution

01

Recall the definition of tangent

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. Mathematically, \(\tan \theta = \frac{\text{opposite}}{\text{adjacent}}\).
02

Identify the special triangle

Consider the 30-60-90 triangle. In this triangle, the sides have a specific ratio: the side opposite the 30掳 angle is 1, the side opposite the 60掳 angle is \(\tan(60^{\text{circ}}) = 1 \).
03

Apply the ratio to \(\tan 60^{\text{circ}}\)

In a 30-60-90 triangle, the tangent of 60掳 can be calculated using the side lengths. Since the opposite side to 60掳 is \(\sqrt{3}\)\times \text{adjacent}, \(\tan(60掳) = \frac{\sqrt{3}}{1} = \sqrt{3}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

tangent function
The tangent function, often denoted as \tan\theta, is a fundamental trigonometric function. In a right triangle, the tangent of an angle \(\theta\) is defined as the ratio of the length of the opposite side to the length of the adjacent side.
This relationship is written mathematically as \(\tan\theta = \frac{\text{opposite}}{\text{adjacent}}\).

Understanding the tangent function helps in finding the relationships between different angles and side lengths in right triangles. It's part of the trio of primary trigonometric functions, including sine and cosine. Knowing one function's value often helps to find the values of the others.
special triangles
Special triangles are right triangles with certain predictable relationships among their angles and sides. They help simplify solving trigonometric problems.
Two primary types of special triangles are:
  • 30-60-90 triangles
  • 45-45-90 triangles

In these triangles, the angles and side proportions always follow specific patterns.

For example, in a 45-45-90 triangle, the legs are equal, and the hypotenuse is \(\sqrt{2}\) times the length of one leg. These properties make it easier to calculate trigonometric functions without needing a calculator.
30-60-90 triangle
A 30-60-90 triangle is a type of special triangle with angles of 30 degrees, 60 degrees, and 90 degrees. The sides of this triangle have a predictable ratio.
The side opposite the 30-degree angle is half the hypotenuse. The side opposite the 60-degree angle is \(\sqrt{3}\) times the side opposite the 30-degree angle.
This can be summarized as:
  • The side opposite 30掳 = 1
  • The side opposite 60掳 = \(\sqrt{3}\)
  • The hypotenuse = 2

To find \(\tan 60^{\circ}\), use the ratio \(\frac{\text{opposite to 60掳}}{\text{adjacent to 60掳}}\). In this case, it is \(\frac{\sqrt{3}}{1} = \sqrt{3}\). Knowing these relationships allows you to quickly determine the tangent and other trigonometric functions in 30-60-90 triangles.

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