Problem 120 - Precalculus

Question

For the following exercises, evaluate the expressions. $$ \tan ^{-1}(\sqrt{3}) $$

Step-by-Step Solution

Answer

The value of \( \tan^{-1}(\sqrt{3}) \) is \( \frac{\pi}{3} \) radians.

Step 1: Understand the Problem

You are asked to find \( \tan^{-1}(\sqrt{3}) \). This means you need to find an angle \( \theta \) such that \( \tan(\theta) = \sqrt{3} \).

Step 2: Recall Tangent Values

Recall the common values of tangent for special angles. The tangent function \( \tan(\theta) \) is known to equal \( \sqrt{3} \) at the angle \( \theta = \frac{\pi}{3} \) radians, which is equivalent to \( 60^{\circ} \).

Step 3: Use the Inverse Function

Since \( \tan^{-1}(x) \) is the inverse of the tangent function and returns values in the range \[ -\frac{\pi}{2}, \frac{\pi}{2} \], \( \tan^{-1}(\sqrt{3}) \) will return \( \frac{\pi}{3} \) radians.

Step 4: Confirm the Solution

Ensure the angle found matches the condition. At \( \theta = \frac{\pi}{3} \), the tangent value is indeed \( \tan(\frac{\pi}{3}) = \sqrt{3} \), so the solution is correct.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are essential tools in trigonometry, allowing us to determine angles based on given trigonometric ratios. When you see a function like \( \tan^{-1}(x) \), it asks the question: for what angle \( \theta \) is \( \tan(\theta) = x \)? These functions, including \( \sin^{-1} \), \( \cos^{-1} \), and \( \tan^{-1} \), effectively "undo" the trigonometric functions themselves. The inverse functions return angles as their output, specifically in certain restricted ranges. For \( \tan^{-1}(x) \), the output angle \( \theta \) is always between \(-\frac{\pi}{2}\) and \( \frac{\pi}{2}\) radians (or \(-90^{\circ}\) and \(90^{\circ}\)). These restricted ranges are necessary because trigonometric functions are periodic and can repeat values, so their inverses need specific constraints to provide unique outputs. Applying \( \tan^{-1} \) helps find the angle if you know the tangent's value, just as in the exercise where evaluating \( \tan^{-1}(\sqrt{3}) \) leads to finding the specific angle \( \theta \).

Tangent Function

The tangent function is one of the primary trigonometric functions, alongside sine and cosine. It is defined as the ratio of the opposite side to the adjacent side in a right-angled triangle. This function can be described mathematically as \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \), which means it involves both sine and cosine.In different contexts:

• The tangent of an angle in the unit circle is the y-coordinate divided by the x-coordinate of the corresponding point on the circle.
• Tangent is periodic with a period of \( \pi \) radians (or \(180^{\circ}\)). This means it repeats its values every \( \pi \) radians.
• Tangent can become undefined when the cosine of the angle is zero, leading to vertical asymptotes in its graph.

For special angles, like those used in the exercise, the tangent values are often known by heart or calculated using basic geometry. For instance, at \( \frac{\pi}{3} \) radians (\(60^{\circ}\)), the tangent value is \( \sqrt{3} \). Knowing these "key" angles can simplify solving many trigonometric problems.

Special Angles

Special angles in trigonometry refer to specific angles that have well-known trigonometric ratio values. These include angles like \(30^{\circ} \), \(45^{\circ} \), and \(60^{\circ} \), or in radians, \( \frac{\pi}{6}, \frac{\pi}{4}, \) and \( \frac{\pi}{3} \), respectively.These angles are special because:

• They often result in trigonometric ratios of simple, calculable, and repeatable values like \( \frac{1}{2}, \sqrt{2}/2, \) or \( \sqrt{3}/2 \).
• Each special angle can be derived from a specific triangle, such as an equilateral triangle or a right isosceles triangle, ensuring precise tangent, sine, and cosine values without a calculator.
• They form the foundation for more complex trigonometric calculations, making them invaluable in both theoretical problems and practical applications, such as physics and engineering.

Understanding these angles, like how \( \tan(\frac{\pi}{3}) = \sqrt{3} \), allows us to swiftly solve problems like finding \( \tan^{-1}(\sqrt{3}) \), which directly relates to these special angles. By remembering these fundamental angles and their trigonometric values, you enhance your skills for both academic exercises and real-world applications.