Problem 40 - Precalculus Essentials

Question

Find the exact value of each expression, if possible. Do not use a calculator. $$\tan ^{-1}\left[\tan \left(-\frac{\pi}{3}\right)\right]$$

Step-by-Step Solution

Answer

The exact value of the expression \(\tan^{-1}[\tan(-\pi/3)]\) is \(-\pi/3\).

Step 1: Evaluate the expression inside the inverse tangent

First, we need to find out what \(\tan (-\pi/3)\) equals to. Considering the periodic nature of the tangent function, \(\tan(\theta) = \tan (\theta + n\pi)\) for every integer \(n\), and that tangent of an angle is the opposite of that negative angle, it follows that \(\tan (-\pi/3) = -\tan(\pi/3)\). Knowing that \(\tan(\pi/3) = \sqrt{3}\), we therefore find \(\tan (-\pi/3) = -\sqrt{3}\).

Step 2: Substitute the evaluated tangent into the inverse tangent expression

Having evaluated \(\tan (-\pi/3)\), we substitute this value back into the original expression: \(\tan^{-1}[ \tan(-\pi/3)] = \tan^{-1}(-\sqrt{3})\).

Step 3: Determine the principal value of the inverse tangent

The function \(\tan^{-1}(x)\) gives an angle whose tangent is \(x\). Thus, \(\tan^{-1}(-\sqrt{3})\) gives the angle in the interval \(-\pi/2\) to \(+\pi/2\) whose tangent is \(-\sqrt{3}\). Looking at the values of tangent, we know that \(\tan(-\pi/3) = -\sqrt{3}\). The principal value of the inverse tangent is therefore \(-\pi/3\).

Key Concepts

Understanding the Tangent Function

The tangent function, often denoted as \( \tan(\theta) \), is one of the primary trigonometric functions. It relates the angle \( \theta \) in a right triangle to the ratio of the side opposite the angle to the adjacent side.
A few key points about tangent:

• It is defined as \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \).
• The value of tangent can be any real number because it is not limited like sine or cosine.
• Critical angles at which the tangent is defined include 0, \( \frac{\pi}{4} \), \( \frac{\pi}{2} \) (undefined), and corresponding negative values.

For students new to this concept, it might be useful to understand how the tangent function behaves with respect to certain angles. For example, at \( \theta = \frac{\pi}{3} \), the tangent value is \( \sqrt{3} \) because it corresponds to a 60-degree angle in a special right triangle where the sides follow a 1: \( \sqrt{3} \): 2 ratio.
This function can also handle negative angles, where, \( \tan(-\theta) = -\tan(\theta) \), demonstrating symmetry about the origin.

Grasping the Principal Value of Inverse Tangent

The principal value of the inverse tangent, represented as \( \tan^{-1}(x) \), is one of the essential concepts in understanding inverse trigonometric functions. Inverse tangent takes a real number \(x\) and returns an angle \(\theta\) such that \( \tan(\theta) = x \).
Key points about the principal value are:

• The range of \( \tan^{-1}(x) \) is restricted to \(-\frac{\pi}{2}, \frac{\pi}{2} \). This ensures that each real number \(x\) corresponds to a unique angle \(\theta\).
• The principal value is needed because \( \tan(\theta) \) is a periodic function. It repeats its values at intervals of \( \pi \).

In the context of the given problem, when finding \( \tan^{-1}(-\sqrt{3})\), one seeks the angle that results in \(-\sqrt{3}\) when its tangent is taken. For this exercise, that angle is \(-\frac{\pi}{3}\), fitting within the defined range.

Exploring the Periodicity of Tangent

The periodicity of the tangent function is a defining feature that affects how we compute its values and their inverses. When we say the tangent function is periodic, it means that it repeats its values in a regular, cyclical pattern over intervals of \( \pi \) radians.
Here is what you need to know about its periodic nature:

• For any angle \( \theta \), \( \tan(\theta) = \tan(\theta + n\pi) \) for any integer \( n \). This means the tangent function repeats every \( \pi \) radians.
• Its periodicity leads to the function having a vertical asymptote at \( \theta = \frac{\pi}{2} \) and \( \theta = -\frac{\pi}{2} \), where it becomes undefined.

Understanding this repetition allows us to find angles with equivalent tangent values, crucial when working through inverse trigonometric problems. In practice, when evaluating \( \tan(-\pi/3) \), knowing the periodicity helps to quickly move to equivalent angles within the principal range of \( \tan^{-1} \). Hence, recognizing that \( \tan(\theta) = \tan(-\pi/3) = -\sqrt{3} \) leads directly to solving for the principal value angle.