Question

Use a coterminal angle to find the exact value of each expression. Do not use a calculator. $$ \tan 390^{\circ} $$

Short answer

\( \tan 390^{\text{°}} = \frac{\sqrt{3}}{3} \)

Step-by-step solution

Understand Coterminal Angles

Coterminal angles are angles that share the same terminal side. They differ by full rotations of 360 degrees. Therefore, adding or subtracting multiples of 360 degrees to/from an angle will give you a coterminal angle.

Find a Coterminal Angle

To find a coterminal angle for \(390^{\text{°}}\), subtract 360 degrees because 390 is greater than 360. This gives: \(390^{\text{°}} - 360^{\text{°}} = 30^{\text{°}}\).

Use the Tangent Function

Once the coterminal angle is found, which is \(30^{\text{°}}\), recognize that \(\tan 30^{\text{°}} = \frac{1}{\sqrt{3}}\). This value is typically simplified to \(\frac{\sqrt{3}}{3}\).

State the Exact Value

Now state the exact value using the information from the previous step. Thus, \( \tan 390^{\text{°}} = \frac{\sqrt{3}}{3} \).

Key concepts

tangent function

To fully understand the solution to the exercise, it's important to grasp what the tangent function is. The tangent function, often written as \(\tan\), is one of the primary trigonometric functions. In a right-angled triangle, it relates the angle to the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. Mathematically, this is expressed as:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]
The tangent function is periodic, which means it repeats its values in regular intervals of 180 degrees or \( \frac{\text{π}}{1} \text{ radians} \). This property is very useful when dealing with angles greater than 360 degrees or less than 0 degrees. Understanding the periodic nature of the tangent function helps in finding equivalent angles, which leads us to the concept of coterminal angles.

trigonometry

Trigonometry is a branch of mathematics focused on studying the relationships between angles and sides of triangles, particularly right triangles. It deals with trigonometric functions like sine (\(\text{sin}\)), cosine (\(\text{cos}\)), and tangent (\(\text{tan}\)). These functions help in solving problems related to angles and distances in various fields such as physics, engineering, and computer science. It’s essential to understand trigonometric functions to solve the given problem effectively.
Here are some key points about trigonometry:

• They relate the angles of a triangle to the lengths of its sides.
• They can be used to describe rotations and periodic phenomena.
• You can use trigonometric identities to simplify complex expressions.

In the provided exercise, we used the trigonometric function tangent (\(\text{tan}\)) and the concept of coterminal angles to find the exact value.

exact values

When we talk about exact values in trigonometry, we refer to the precise values of trigonometric functions at specific angles without using a calculator. Using exact values is vital for accuracy in mathematical problems and proofs. Knowing the exact values of trigonometric functions for common angles like 30°, 45°, and 60° is particularly helpful.
For instance:

• \(\tan 30^{\text{°}} = \frac{\text{opposite}}{\text{adjacent}} = \frac{1}{\text{√3}}\), simplified to \(\frac{\text{√3}}{3} \).
• \(\tan 45^{\text{°}} = 1\)
• \(\tan 60^{\text{°}} = \text{√3}\)

In the given exercise, the exact value of \(\tan 390^{\text{°}}\) was found by first determining its coterminal angle (30°) and then using the known exact value of \(\tan 30^{\text{°}} = \frac{\text{√3}}{3}\).
Understanding these exact values and how to derive them is crucial for solving a wide range of trigonometric problems.