Problem 212 - Calculus Volume 1

Question

Evaluate the functions. Give the exact value. \(\cos \left(\tan ^{-1}(\sqrt{3})\right)\)

Step-by-Step Solution

Answer

The exact value is \( \frac{1}{2} \).

Step 1: Understand the Inverse Tangent Function

The expression involves \( \tan^{-1}(\sqrt{3}) \). This function asks for the angle \( \theta \) such that \( \tan(\theta) = \sqrt{3} \).

Step 2: Identify the Angle

From trigonometric values, we know that \( \tan(\pi/3) = \sqrt{3} \). Thus, \( \tan^{-1}(\sqrt{3}) = \pi/3 \).

Step 3: Substitute and Simplify

Now, substitute the angle into the cosine function: \( \cos(\tan^{-1}(\sqrt{3})) = \cos(\pi/3) \).

Step 4: Evaluate the Cosine Function

Recall that \( \cos(\pi/3) = 1/2 \). Thus, \( \cos(\tan^{-1}(\sqrt{3})) = 1/2 \).

Key Concepts

Understanding Inverse Tangent

The inverse tangent function, often denoted as \( \tan^{-1}(x) \) or "arctan," helps us find an angle \( \theta \) when the tangent of that angle is known. This function essentially reverses the tangent function. In our exercise, we have \( \tan^{-1}(\sqrt{3}) \). This means we're looking for an angle where its tangent value is \( \sqrt{3} \). This will often relate back to what we call "special angles," which are angles whose trigonometric values are well-known.

• In trigonometry, the tangent of \( \pi/3 \) is \( \sqrt{3} \).
• Thus, \( \tan^{-1}(\sqrt{3}) \) is identical to \( \pi/3 \), meaning the angle \( \theta \) is \( \pi/3 \).

Understanding these relationships helps us make connections between angles and their trigonometric functions effectively.

The Role of the Cosine Function

The cosine function is crucial for evaluating the trigonometric expression \( \cos(\tan^{-1}(\sqrt{3})) \). The cosine of an angle, represented as \( \cos(\theta) \), relates to the adjacent side over the hypotenuse in a right triangle. Once we've identified \( \theta \) from the \( \tan^{-1} \) function, we can use this angle within the cosine function.

• From earlier, we found that \( \tan^{-1}(\sqrt{3}) = \pi/3 \).
• The cosine of \( \pi/3 \) is a well-known trigonometric value, which is \( \frac{1}{2} \).

Hence, upon substituting it into the cosine function, we find \( \cos(\tan^{-1}(\sqrt{3})) = \cos(\pi/3) = \frac{1}{2} \). Recognizing and memorizing these values can simplify trigonometry problems considerably.

Common Trigonometric Values and Angles

Understanding the specific values of trigonometric functions for certain angles is very helpful in trigonometry. These special angles, such as \( \pi/6 \), \( \pi/4 \), and \( \pi/3 \), have trigonometric values that make calculations straightforward. For example,

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

These values are not arbitrary; they come from the properties of equilateral and special right triangles.
Remembering these trigonometric values allows you to solve problems more easily and recognize patterns quickly. They serve as a key foundation in understanding more complex trigonometric identities and equations.