Problem 18 - Algebra and Trigonometry

Question

Find the exact value of each expression. $$ \tan ^{-1}\left(-\frac{\sqrt{3}}{3}\right) $$

Step-by-Step Solution

Answer

-\frac{\pi}{6}

Step 1: Understanding the \(\tan^{-1}(x)\) function

The inverse tangent function, denoted \(\tan^{-1}(x)\) or arctan(x), is the inverse of the tangent function. It returns the angle whose tangent is `x`. The output of an inverse tangent function is an angle in the interval \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\) (or \(-90^\circ\) to \(90^\circ\)).

Step 2: Knowing the special values of tangent

Important values of tangent for standard angles are: \(\tan{\frac{\pi}{6}} = \frac{\sqrt{3}}{3}\), \(\tan{\frac{\pi}{4}} = 1\), and \(\tan{\frac{\pi}{3}} = \sqrt{3}\). In this case, we are asked to find \(\tan^{-1}\left(-\frac{\sqrt{3}}{3}\right)\), the angle whose tangent is \(-\frac{\sqrt{3}}{3}\). This corresponds to \(-\frac{\pi}{6}\) for positive, but because we're given a negative value, the angle resides in the fourth quadrant. Therefore, this angle is \(-30^\circ\) or \(-\frac{\pi}{6}\).

Step 3: Result

So, \(\tan^{-1}\left(-\frac{\sqrt{3}}{3}\right) = -\frac{\pi}{6}\)

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental in understanding angles and relationships in triangles. They include sine, cosine, and tangent, which are the primary ones used in various calculations. For any given angle, these functions can help determine the ratios between the sides of a right triangle. When we talk about the inverse of these functions, they help us find an angle with a known ratio.

• Sine (\(\sin\theta\)): Ratio of the opposite side to the hypotenuse.
• Cosine (\(\cos\theta\)): Ratio of the adjacent side to the hypotenuse.
• Tangent (\(\tan\theta\)): Ratio of the opposite side to the adjacent side.

For the inverse functions, such as \(\tan^{-1}(x)\), we aim to find the angle whose tangent is \(x\). This is crucial in problems where the angle, not the ratio, is unknown.

Special Angles in Trigonometry

Special angles in trigonometry are angles that have well-known sine, cosine, and tangent values. These include \(30^\circ\), \(45^\circ\), and \(60^\circ\). Recognizing these angles can aid in solving trigonometric problems quickly.

• \(30^\circ\) or \(\frac{\pi}{6}\): \(\tan\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{3}\)
• \(45^\circ\) or \(\frac{\pi}{4}\): \(\tan\left(\frac{\pi}{4}\right) = 1\)
• \(60^\circ\) or \(\frac{\pi}{3}\): \(\tan\left(\frac{\pi}{3}\right) = \sqrt{3}\)

These values are often memorized to make calculations faster. When working with negative angles or inverse functions, knowing these special angles helps us determine the resulting angle accurately.

Unit Circle

The unit circle is a circle with a radius of 1, centered at the origin of a coordinate plane. It is a powerful tool in trigonometry for finding angles and their corresponding function values.

The coordinates of any point on the unit circle correspond to the cosine and sine of that angle, i.e., \((\cos\theta, \sin\theta)\). This helps visualize and remember trigonometric ratios easily. For tangent values, you can use the sine and cosine to compute the ratios:

• \(\tan\theta = \frac{\sin\theta}{\cos\theta}\)

By knowing these relationships, finding values like \(\tan^{-1}\left(-\frac{\sqrt{3}}{3}\right)\) becomes straightforward. The unit circle helps identify angles, including those in different quadrants, based on given trigonometric values.

Negative Angles

Negative angles in trigonometry are simply angles measured in the clockwise direction from the positive x-axis. In trigonometric functions, negative angles often correspond to points in other quadrants of the coordinate plane.

For example, \(-30^\circ\) is equivalent to rotating 30 degrees clockwise, placing it in the fourth quadrant. This makes it vital to understand how the signs of trigonometric functions change with negative angles:

• In the fourth quadrant, sine is negative, cosine is positive, and tangent is negative.

Thus, knowing how to work with negative angles can help recognize \(\tan^{-1}\left(-\frac{\sqrt{3}}{3}\right)\) as \(-\frac{\pi}{6}\), by using the property that only the angle direction differs while its trigonometric value stays the same.