Problem 245 - Precalculus

Question

For the following exercises, find the exact value. $$ \tan ^{-1}(\sqrt{3}) $$

Step-by-Step Solution

Answer

\( \frac{\pi}{3} \)

Step 1: Understand the Inverse Function

The function \( \tan^{-1}(x) \) is the inverse of the tangent function, thus it gives us an angle \( \theta \) such that \( \tan(\theta) = x \). In this problem, we need to find \( \theta \) for which \( \tan(\theta) = \sqrt{3} \).

Step 2: Recall Tangent Values for Special Angles

Recall that \( \tan(\theta) = \sqrt{3} \) when \( \theta \) corresponds to one of the special angles on the unit circle. The standard angle with this tangent value is \( \frac{\pi}{3} \) (or 60 degrees) because \( \tan\left(\frac{\pi}{3}\right) = \frac{\sin\left(\frac{\pi}{3}\right)}{\cos\left(\frac{\pi}{3}\right)} = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \sqrt{3}\).

Step 3: Determine the Range of \( \tan^{-1}(x) \)

The range of \( \tan^{-1}(x) \) is \((-\frac{\pi}{2}, \frac{\pi}{2})\). The angle \( \frac{\pi}{3} \), which is approximately 1.047 radians, falls within this range. Therefore, it is a valid solution.

Step 4: Conclude the Solution

Since the only angle within the range \((-\frac{\pi}{2}, \frac{\pi}{2})\) that satisfies \( \tan(\theta) = \sqrt{3} \) is \( \theta = \frac{\pi}{3} \), the exact value of \( \tan^{-1}(\sqrt{3}) \) is \( \frac{\pi}{3} \).

Key Concepts

Tangent Function

The tangent function is a fundamental trigonometric function that often plays a critical role in various mathematical contexts. It relates to the ratio of the opposite side to the adjacent side in a right triangle. In terms of the unit circle, it is the ratio of the y-coordinate (sine) to the x-coordinate (cosine).

• Its representation is given by \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \).
• This means when either sine or cosine equals zero, the function reaches infinity, leading to asymptotes at these points.
• Unlike sine and cosine, the tangent function is periodic over \( \pi \) radians (or 180 degrees) instead of \( 2\pi \).

The inverse of the tangent function, \( \tan^{-1}(x) \), is used to find an angle whose tangent is \( x \). It provides solutions within the range \( (-\frac{\pi}{2}, \frac{\pi}{2}) \). This range ensures that for every \( x \), there is a unique angle \( \theta \) which satisfies the condition \( \tan(\theta) = x \).

Special Angles

Special angles are key to quickly solving trigonometric problems without the need for a calculator. They are standardized angles whose sine, cosine, and tangent values are well-known.

• The basic special angles are \( 0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2} \), equivalent to 0, 30, 45, 60, and 90 degrees, respectively.
• For these angles, trigonometric ratios such as \( \tan(\frac{\pi}{3}) = \sqrt{3} \) are memorized.
• These values are derived from specific right triangles like the 30-60-90 and 45-45-90 triangles, which provide consistent ratios among their sides.

Understanding these angles helps solve equations like \( \tan^{-1}(\sqrt{3}) \), as you can immediately recognize the corresponding angle \( \theta \) where this condition is satisfied.

Unit Circle

The unit circle is a circle of radius one, centered at the origin of a coordinate plane. It is a powerful tool in trigonometry since it defines trigonometric functions for all real numbers and helps visualize their behavior across the angles.

• Points on the unit circle are expressed in terms of their coordinates, \( (\cos(\theta), \sin(\theta)) \).
• These coordinates correspond to the angle formed with the positive x-axis.
• Tangent values can be found by the slope created by this angle from the origin to the circle's edge, which is \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \).
• The specific quadrant of the angle can affect the sign and value of these functions.

Using the unit circle, you can confirm trigonometric identities, compute various trigonometric values, and solve problems like \( \tan^{-1}(\sqrt{3}) \) by identifying the corresponding angle at which the tangent equals \( \sqrt{3} \), which is \( \frac{\pi}{3} \). It serves as a reference for many trigonometric calculations and understanding of function behavior.