Problem 212 - Calculus Volume 1

Question

For the following exercises, evaluate the functions. Give the exact value. $$ \cos \left(\tan ^{-1}(\sqrt{3})\right) $$

Step-by-Step Solution

Answer

\( \frac{1}{2} \)

Step 1: Understand the Inverse Trigonometric Function

The problem involves evaluating the cosine of an angle whose tangent is \( \sqrt{3} \). In other words, we need an angle \( \theta \) such that \( \tan(\theta) = \sqrt{3} \). This implies finding \( \theta = \tan^{-1}(\sqrt{3}) \).

Step 2: Identify the Relevant Angle

Recall that \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \). If \( \tan(\theta) = \sqrt{3} \), it can also be written as \( \tan(\theta) = \frac{\sqrt{3}}{1} \). This ratio matches the tangent of a 60-degree or \( \frac{\pi}{3} \) radian angle in a right triangle.

Step 3: Determine the Cosine of the Angle

Now, since \( \theta = \frac{\pi}{3} \), we need to find \( \cos(\theta) \). For a 60-degree angle or \( \frac{\pi}{3} \) radians, \( \cos(\theta) = \frac{1}{2} \).

Step 4: Evaluate and Verify

Thus, \( \cos(\tan^{-1}(\sqrt{3})) = \cos(\frac{\pi}{3}) = \frac{1}{2} \). Verify this by checking standard trigonometric values for common angles.

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental tools in understanding and simplifying expressions involving trigonometric functions. They are equations that hold true for all values of the involved variables. In the context of inverse trigonometric functions, knowing the right identities can simplify the process of evaluating complex expressions. For example, an important identity relates the tangent and cosine of an angle:

• For any angle \( \theta \), \( \tan(\theta) \) can be expressed as the ratio \( \frac{\text{opposite}}{\text{adjacent}} \).
• Using the Pythagorean identity, \( \sin^2(\theta) + \cos^2(\theta) = 1 \), helps find one function if the other is known.

Understanding how inverse functions like \( \tan^{-1}\) and\( \cos \) are used together can assist in arriving at exact values in trigonometric evaluations.

Angle Evaluation

Angle evaluation involves determining the measure of an angle that satisfies certain conditions, often given in terms of trigonometric functions. In the exercise, we needed the angle \( \theta \) such that \( \tan(\theta) = \sqrt{3} \). This involves remembering common angle measures:

• When evaluating \( \tan^{-1}(\sqrt{3}) \), it translates into: Find an angle \( \theta \) that has a tangent of \( \sqrt{3} \).
• This value corresponds to a specific angle on the unit circle, where \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{\sqrt{3}}{1} \), which is equivalent to \( 60^\circ \) or \( \frac{\pi}{3} \) radians.

Angle evaluation is crucial when working with inverse functions since it allows for finding exact solutions based on known trigonometric values.

Exact Values of Trigonometric Functions

Exact values of trigonometric functions are often derived from special angles used in mathematics. Angles such as \( 30^\circ, 45^\circ, \) and \( 60^\circ \) have specific sine, cosine, and tangent values. These are often memorized or quickly referenced. For example:

• For \( 60^\circ \) or \( \frac{\pi}{3} \), \( \cos(\theta) \) is known to be \( \frac{1}{2} \).
• These values are derived from a 30-60-90 triangle, a special right triangle, where the sides are in the ratio 1: \( \sqrt{3}: 2 \).

In evaluating expressions like \( \cos(\tan^{-1}(\sqrt{3})) \), knowing these exact trigonometric values is essential. This knowledge allows students to skip lengthy calculations and arrive at solutions more efficiently.