Problem 119 - Precalculus

Question

For the following exercises, evaluate the expressions. $$ \tan ^{-1}(-1) $$

Step-by-Step Solution

Answer

\( \tan^{-1}(-1) = -\frac{\pi}{4} \).

Step 1: Understanding the Inverse Tangent Function

The inverse tangent function, denoted as \( \tan^{-1}(x) \), finds the angle \( \theta \) whose tangent is \( x \). Hence, \( \tan(\theta) = x \). The range of \( \tan^{-1}(x) \) is typically \((-\frac{\pi}{2}, \frac{\pi}{2})\).

Step 2: Setting Up the Equation

Since we are looking for \( \tan^{-1}(-1) \), we need to find the angle \( \theta \) such that \( \tan(\theta) = -1 \) and \( \theta \) is within the range \((-\frac{\pi}{2}, \frac{\pi}{2})\).

Step 3: Identifying the Reference Angle

The angle whose tangent is 1 is \( \frac{\pi}{4} \). However, since we need the tangent to be \(-1\), we consider the angle in the fourth quadrant where tangent is negative.

Step 4: Determining the Correct Angle

In the domain \((-\frac{\pi}{2}, \frac{\pi}{2})\), the correct angle is \( -\frac{\pi}{4} \), because \( \tan(-\frac{\pi}{4}) = -1 \).

Step 5: Conclusion

The value of \( \tan^{-1}(-1) \) is \( -\frac{\pi}{4} \).

Key Concepts

Trigonometric Functions

Trigonometric functions are essential in mathematics, especially when dealing with angles in right-angled triangles and periodic phenomena. The primary functions include sine, cosine, and tangent, each relating an angle to different ratios of a right triangle's sides.

The tangent function, denoted as \( \tan(\theta) \), is the ratio of the opposite side to the adjacent side in a right triangle. This function is periodic, meaning it repeats its values over specific intervals.

When we talk about the inverse tangent, denoted as \( \tan^{-1}(x) \), we mean finding the angle \( \theta \) for which \( \tan(\theta) = x \). It helps in determining an angle given a tangent value. The range for this inverse function typically lies between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\), ensuring we have a unique output for each input. This range is crucial because it avoids the repetitive values that arise in the typical periodic nature of the tangent function.

Reference Angle

A reference angle is a helpful concept when working with angles, focusing on the simplest form of an angle in standard position. It is always the smallest angle between the terminal side of a given angle and the x-axis.

To find a reference angle, you need to:

• Identify the quadrant where the angle lies.
• Subtract or add the angle to \(\pi\) or \(2\pi\) to bring it to its basic form.

Reference angles make it simple to compute trigonometric functions for any angle based on their positive or negative signs in different quadrants. For instance, if you know \( \tan(\frac{\pi}{4}) = 1 \), and you're looking for where tangent equals \(-1\), you understand \( \frac{\pi}{4} \) shows where \( \tan \) equals 1, thus \( -\frac{\pi}{4} \) in the fourth quadrant gives \( \tan(-\frac{\pi}{4}) = -1 \).

By using reference angles, you can easily manage the signs of values according to their positions in the coordinate plane. This becomes particularly useful when solving equations such as \( \tan^{-1}(-1) = -\frac{\pi}{4} \).

Angle Measurement

Understanding angle measurement is crucial for solving trigonometric problems. Angles can be measured in degrees or radians, with 360 degrees equivalent to \(2\pi\) radians. Radians are often preferred in mathematics because they provide a natural measure tied to the circle's geometry.

In trigonometry, using radians allows for a seamless connection between angles and their trigonometric functions. Each radian represents an arc length on the unit circle, making calculations with angles easier and more intuitive. For example, \(\frac{\pi}{4}\) radians corresponds to a 45-degree angle, and we know from earlier that \(\tan(\frac{\pi}{4}) = 1\).

When calculating inverse trigonometric functions, such as \(\tan^{-1}(x)\), having a consistent understanding of angle measurement helps identify the correct angle. The solution to \(\tan^{-1}(-1)\) is \(-\frac{\pi}{4}\) because, within the inverse function's defined range, this angle satisfies \(\tan(\theta) = -1\). Using angle measurement consistently across various problems ensures accuracy and comprehension in trigonometry.