Problem 11 - A Graphical Approach to Precalculus with Limits

Question

Find the exact value of each real number \(y .\) Do not use a calculator. $$y=\tan ^{-1} 1$$

Step-by-Step Solution

Answer

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

Step 1: Understand the Inverse Tangent Function

The inverse tangent function, denoted as \( \tan^{-1}(x) \), represents the angle \( \theta \) whose tangent is \( x \). That is, if \( y = \tan^{-1}(1) \), then \( \tan(y) = 1 \). The function takes a real number \( x \) and returns an angle \( y \) in radians.

Step 2: Recall the Special Angles for Tangent

Recall that the tangent of an angle \( \theta \) is equal to \( 1 \) when \( \theta \) is \( \frac{\pi}{4} \) radians. This is because \( \tan\left(\frac{\pi}{4}\right) = 1 \).

Step 3: Determine the Principal Value of \( y \)

The principal values for the inverse tangent function lie in the interval \( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \). Since \( \frac{\pi}{4} \) is within this range, it is valid to say \( y = \frac{\pi}{4} \).

Step 4: Conclude the Exact Value

The exact value of \( y = \tan^{-1}(1) \) is \( \frac{\pi}{4} \). Consequently, \( y = \frac{\pi}{4} \).

Key Concepts

Tangent Function

The tangent function, often written as \( \tan(\theta) \), is a fundamental trigonometric function that relates an angle in a right triangle to the ratio of the opposite side to the adjacent side. When dealing with the tangent function, especially in the context of inverse trigonometry, we often look for angles given a specific tangent value.

In this exercise, we explore the inverse tangent function, noted as \( \tan^{-1}(x) \) or \( \arctan(x) \). This function finds an angle when you know the tangent value. Importantly, the result is typically expressed in radians, where the angle is within the principal range of \((-\frac{\pi}{2}, \frac{\pi}{2})\).

Here's what happens in this exercise: if \( y = \tan^{-1}(1) \), then we are essentially looking for an angle \( y \) where \( \tan(y) = 1 \). Understanding these properties of tangent will help solve more complex trigonometric problems.

Radian Measure

Radian measure is one of the ways to quantify angles; it's especially useful in calculus and higher math due to its natural relationship with arc length and the unit circle.To understand radians, imagine a circle centered at the origin in the coordinate plane, with radius 1. The central angle that intercepts an arc equal in length to the radius of the circle is defined as 1 radian.In our problem, we need to express the angle resulting from \( \tan^{-1}(1) \) in radians. It turns out that this specific angle is \( \frac{\pi}{4} \) radians. This is equivalent to 45 degrees in terms of the typical degree measure. Knowing that \( \pi \) radians equal 180 degrees helps convert and understand these angle measures easily.

Special Angles

Special angles in trigonometry are angles for which the trigonometric functions' values are well-known and easy to calculate without a calculator. Often, these angles are multiples of 30°, 45°, and 60°, or in radian measures, \( \frac{\pi}{6} \), \( \frac{\pi}{4} \), and \( \frac{\pi}{3} \) respectively.In this problem, we recall that one of the special angles is \( \frac{\pi}{4} \), for which \( \tan\left(\frac{\pi}{4}\right) = 1 \). This knowledge allows us to easily find that \( \tan^{-1}(1) = \frac{\pi}{4} \), demonstrating the significance of these angles.These special cases are frequently encountered in trigonometric problems, so remembering these values is crucial for solving them quickly and accurately.