Problem 197 - Precalculus

Question

For the following exercises, find the exact value without the aid of a calculator. $$ \tan ^{-1}(-1) $$

Step-by-Step Solution

Answer

The exact value is \( -\frac{\pi}{4} \).

Step 1: Understand Inverse Trigonometric Function

The function \( \tan^{-1}(x) \) is the inverse of the tangent function. It returns the angle \( \theta \) whose tangent is \( x \). For \( \tan^{-1}(-1) \), we need to find an angle such that \( \tan(\theta) = -1 \).

Step 2: Consider the Range of \( \tan^{-1}(x) \)

The range of the \( \tan^{-1}(x) \) function is \((-\frac{\pi}{2}, \frac{\pi}{2})\). This means the angle \( \theta \) must be between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\).

Step 3: Recall Tangent Values for Common Angles

Recall that \( \tan\left(-\frac{\pi}{4}\right) = -1 \). This is because the tangent of \(-\frac{\pi}{4}\) is the same as the opposite over the adjacent in a right triangle, which is \(-1\).

Step 4: Verify the Angle

Verify that \( \theta = -\frac{\pi}{4} \) is within the range \((-\frac{\pi}{2}, \frac{\pi}{2})\). Since \(-\frac{\pi}{4}\) is indeed in this range, it is the correct solution.

Key Concepts

Arctangent

The arctangent, often denoted as \( \tan^{-1}(x) \), is an inverse trigonometric function, crucial to understanding angles from tangent values. Unlike the basic tangent function, which computes the ratio of the opposite to adjacent sides in a right triangle given an angle, the arctangent takes a ratio as input and returns the corresponding angle. This angle, \( \theta \), is such that the tangent of \( \theta \) equals the input value. For our exercise, we need to find an angle with a tangent of \(-1\). This is a classic application of the arctangent, pointing out the necessity of identifying the angle itself.

• Arctangent returns angles from a specific range, usually between \( -\frac{\pi}{2} \) and \( \frac{\pi}{2} \).
• For negative input values like \(-1\), the outcome remains in this range, providing angles that yield negative tangent values.

Understanding the arctangent is key, as it helps reverse the process of finding the tangent, offering the corresponding angle as the solution.

Tangent Function

The tangent function, described by \( \tan(\theta) \), is a fundamental element of trigonometry. It represents the ratio of the opposite side to the adjacent side of a right-angled triangle. This function is periodic, repeating its values over intervals. Importantly, it has certain values it commonly reaches at specific angles. For instance, the tangent of \( \theta = -\frac{\pi}{4} \) is \(-1\).

• This periodic nature means the same\(-1\) value appears at various angles, but the inverse function, such as arctangent, picks the principal value within its range.
• Tangent values can tell us a lot about complementary angles and coterminal angles within the unit circle.

Appreciating the behavior of the tangent function and where it is typically negative or positive helps in solving equations involving inverse trigonometric functions like arctangent.

Trigonometric Values

Trigonometric values define the key relationships between angles and side ratios in triangles. They are especially useful when working backward from known values, as in inverse trig functions. When asked to find \( \tan^{-1}(-1) \), knowledge of standard tangent values comes into play. Thinking about familiar angles like \( -\frac{\pi}{4} \) immediately leads to the solution since we know \( \tan(-\frac{\pi}{4}) = -1 \).

• Familiarity with key angles and their sine, cosine, and tangent values is crucial. These include \( \frac{\pi}{4} \), \( \frac{\pi}{3} \), and others.
• Using these well-known angles helps in determining the result quickly, ensuring correct mapping within given ranges.

In essence, internalizing common trigonometric values helps us quickly derive or verify solutions in problems related to inverse trigonometric functions.