Problem 19 - Precalculus: A Concise Course

Question

Evaluate the expression without using a calculator. $$ \tan ^{-1} 0 $$

Step-by-Step Solution

Answer

\(\tan^{-1}0 = 0\)

Step 1: Understanding the Inverse Tangent Function

The expression given is the inverse tangent or arctangent of 0, denoted as \(\tan^{-1}0\). It is asking for an angle in the interval \([- \frac{\pi}{2}, \frac{\pi}{2}]\) whose tangent is 0.

Step 2: Evaluating the Inverse Tangent

The tangent of an angle is defined as the ratio of the length of the side opposite to the angle to the length of the adjacent side in a right triangle. The value of the tangent function at 0 degrees or 0 radians is 0, as there is no opposite side \(0/1 = 0\). Therefore, \(\tan^{-1}0 = 0\).

Key Concepts

Understanding Arctangent

Arctangent, often written as \( \tan^{-1} \), is a function that helps us find an angle when we know the value of the tangent of that angle. It is one of six inverse trigonometric functions which includes arcsine and arccosine. The arctangent of a value answers the question: "What angle has this tangent?"

The range of the arctangent function is limited to

• \([-\frac{\pi}{2}, \frac{\pi}{2}]\)

This restriction ensures that each input corresponds to exactly one angle, making the function's output consistent and predictable. Knowing this range is useful when solving problems, as it tells us the angle outputs we should expect.

What is Tangent Function?

The tangent function, \( \tan(\theta) \), is a fundamental trigonometric function. It expresses the relationship between angles and side lengths in a right triangle. To understand it, we must first think of a right triangle, which has one 90-degree angle.

The formula to find the tangent of an angle \( \theta \) is:

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

This formula helps show how large the angle is based on the length of the side opposite to it compared to the side next to it. Understanding this relationship is key to solving many problems in trigonometry.

Basics of Angle Measurement

In trigonometry, angles are often measured in radians and degrees. Understanding these measurement units is essential for evaluating trigonometric functions.

Degrees: A full circle is 360 degrees.

Radians: A complete circle is \(2\pi\) radians.

Both systems are used interchangeably in mathematics, but radians are more common in trigonometry because they simplify many equations. For example, the right angle in a triangle is \(\frac{\pi}{2}\) radians or 90 degrees. It's crucial to be comfortable converting between these units to solve trigonometry problems efficiently.

Understanding Right Triangles

A right triangle is a special type of triangle that has one angle equal to 90 degrees. This property makes right triangles extremely valuable in trigonometry.

When working with right triangles:

• One angle is always 90 degrees.
• The "hypotenuse" is the longest side, opposite the right angle.
• The other two sides are the "adjacent" and "opposite" sides in relation to a given non-right angle.

Because of these unique properties, right triangles provide a basis for defining trigonometric functions like sine, cosine, and tangent. They are widely used in real-world applications, including architecture, navigation, and physics.