Problem 18 - Algebra and Trigonometry

Question

Find the exact value of the expression, if it is defined. \(\tan ^{-1}\left(\tan \frac{\pi}{6}\right)\)

Step-by-Step Solution

Answer

The exact value is \( \frac{\pi}{6} \).

Step 1: Identify the Inner Function

We start by identifying the inner function in the expression \( \tan \frac{\pi}{6} \). We know that the tangent of an angle \( \frac{\pi}{6} \) is a well-known value.

Step 2: Calculate \( \tan \frac{\pi}{6} \)

Using the unit circle or trigonometric values, we calculate that \( \tan \frac{\pi}{6} = \frac{1}{\sqrt{3}} \) or equivalently \( \frac{\sqrt{3}}{3} \).

Step 3: Recognize the Inverse Function

The problem also includes the inverse tangent function, \( \tan^{-1} \). This function takes a value and returns an angle whose tangent is the given value.

Step 4: Apply the Inverse Tangent Function

Apply the inverse tangent function to \( \tan^{-1} \left( \frac{\sqrt{3}}{3} \right) \). By the property of inverse trigonometric functions, since the angle \( \frac{\pi}{6} \) resides within the range \( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \), which is the principal range of \( \tan^{-1} \), we can conclude that \( \tan^{-1}\left(\tan \frac{\pi}{6}\right) = \frac{\pi}{6} \).

Step 5: Verify the Domain

Verify that the angle \( \frac{\pi}{6} \) is within the principal range of \( \tan^{-1} \), which is indeed the case as \( \frac{\pi}{6} \) is between \(-\frac{\pi}{2}\) and \( \frac{\pi}{2}\).

Key Concepts

Tangent Function

The tangent function is a fundamental part of trigonometry that relates the ratio of the opposite side to the adjacent side in a right triangle. It is denoted as \( \tan \theta \), where \( \theta \) is the angle in question. In terms of the unit circle, the tangent function can be expressed as the ratio of the sine to the cosine of angle \( \theta \): \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).
The tangent function has unique properties:

• It is periodic with a period of \( \pi \).
• It has vertical asymptotes at odd multiples of \( \frac{\pi}{2} \).
• It is not defined at angles where the cosine value is zero.

For example, the tangent of \( \frac{\pi}{6} \) (or 30 degrees) is known to be \( \frac{1}{\sqrt{3}} \) or equivalently \( \frac{\sqrt{3}}{3} \), a standard value you often encounter in trigonometric calculations. Understanding these properties is crucial for solving problems involving tangent and its inverse.

Unit Circle

The Unit Circle is a tool that greatly aids the understanding of trigonometric functions. It is a circle with a radius of 1 centered at the origin of a coordinate system. The angle \( \theta \) in the unit circle is measured in radians, where \( 2\pi \) radians represent a full circle.
Some critical aspects of the unit circle related to trigonometry include:

• The x-coordinate represents the cosine of \( \theta \).
• The y-coordinate represents the sine of \( \theta \).
• The tangent of \( \theta \) is the y-coordinate divided by the x-coordinate, or \( \tan \theta = \frac{y}{x} \).

Using the unit circle, one can easily determine the trigonometric values of common angles such as \( \frac{\pi}{6}, \frac{\pi}{4}, \) and \( \frac{\pi}{3} \). For instance, at \( \frac{\pi}{6} \), the coordinates are \( \left( \frac{\sqrt{3}}{2}, \frac{1}{2} \right) \), verifying that the tangent is \( \frac{1}{\sqrt{3}} \). This visual representation simplifies complex calculations and helps verify inverse trigonometric functions.

Principal Range

The principal range of an inverse trigonometric function is the specific interval over which the function is most easily defined and guarantees a unique output. For the inverse tangent function, \( \tan^{-1} \), this range is \( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \). This interval ensures each value of \( \tan^{-1} \) is tied to just one angle.
Knowing the principal range is key because it dictates how angles are interpreted with inverse trigonometric functions:

• For \( \tan^{-1} \), any input value in this range will return an angle whose tangent is that value.
• It allows the inverse function to work properly without ambiguity.
• When working with the expression \( \tan^{-1}(\tan \theta) \), \( \theta \) must be in the principal range for \( \tan^{-1} \) to return \( \theta \) itself.

In problems like finding \( \tan^{-1}(\tan \frac{\pi}{6}) \), checking that \( \frac{\pi}{6} \) is within this range ensures accurate and unique results.

Trigonometric Values

Trigonometric values are key numerical results derived from trigonometric functions, often corresponding to special angles. They are used extensively, from solving equations to understanding angles in the unit circle. These values are memorized for their simplicity and are found on the unit circle for efficiency.
Key trigonometric values include:

• \( \sin \frac{\pi}{6} = \frac{1}{2} \)
• \( \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2} \)
• \( \tan \frac{\pi}{6} = \frac{1}{\sqrt{3}} \) or \( \frac{\sqrt{3}}{3} \)

These are derived using simple geometry from a 30-60-90 triangle or directly from the unit circle.
In addition, these well-known values simplify more complex trigonometric expressions and facilitate the understanding of inverse trigonometric functions. For example, knowing \( \tan \frac{\pi}{6} \) helps verify outcomes when taking inverse tangent.