Problem 28 - Algebra and Trigonometry

Question

Find the exact value of the expression, if it is defined. \(\cos ^{-1}\left(\sqrt{3} \sin \frac{\pi}{6}\right)\)

Step-by-Step Solution

Answer

The exact value is \( \frac{\pi}{6} \).

Step 1: Simplify Inner Trigonometric Expression

First, evaluate the inner trigonometric expression. Given the expression is \( \sqrt{3} \sin \frac{\pi}{6} \). We know that \( \sin \frac{\pi}{6} = \frac{1}{2} \). Therefore, \( \sqrt{3} \sin \frac{\pi}{6} = \sqrt{3} \times \frac{1}{2} = \frac{\sqrt{3}}{2} \).

Step 2: Verify Domain Validity for Inverse Cosine

The domain of \( \cos^{-1}(x) \) is \( -1 \leq x \leq 1 \). Check if \( \frac{\sqrt{3}}{2} \) is within this domain. Since \( \frac{\sqrt{3}}{2} \approx 0.866 \), it is within the required domain of the inverse cosine function.

Step 3: Find Exact Value of the Inverse Cosine

We need to find the exact value of \( \cos^{-1}\left(\frac{\sqrt{3}}{2}\right) \). Since \( \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2} \), it follows that \( \cos^{-1}\left(\frac{\sqrt{3}}{2}\right) = \frac{\pi}{6} \).

Key Concepts

Inverse Cosine

The inverse cosine function, also known as arccosine, is a type of inverse trigonometric function used for finding angles when the cosine value is known. When you encounter the notation \( \cos^{-1}(x) \), it asks, "What angle has a cosine of \( x \)?" This function provides angles within a specific range. Typically, the output is between 0 and \( \pi \) radians. This forms the principal range or principal value of the inverse cosine.

Inverse cosine is essential for translating a trigonometric ratio back into an angle. Engineers, scientists, and mathematicians use it when they have trigonometric values but need the corresponding angles for their calculations.

Domain of Functions

Understanding the domain of a function is crucial. It refers to all the possible inputs that can produce real outputs. When we talk about the domain of the inverse cosine function, \( \cos^{-1}(x) \), we're looking at all the values \( x \) can hold.

For the inverse cosine, its domain is \(-1 \leq x \leq 1\). This means the inverse cosine can only take input values ranging from -1 to 1, inclusive. Inputs outside this domain won't yield real results using the inverse cosine. For instance, if you're trying to find the inverse cosine of a value greater than 1, like 1.5, it's not possible as this doesn't fall within the acceptable range of \([ -1, 1 ]\).

Exact Value Calculations

Calculating the exact value in trigonometric expressions often involves recognizing special angles and their trigonometric ratios. For example, exact values are computed directly from angles like \( \frac{\pi}{3} \), \( \frac{\pi}{4} \), and \( \frac{\pi}{6} \), which correspond to well-known ratios.

In the problem, once simplified, we find \( \cos^{-1}\left(\frac{\sqrt{3}}{2}\right) \). Knowing that the cosine of \( \frac{\pi}{6} \) is \( \frac{\sqrt{3}}{2} \) helps evaluate the expression quickly.

• This "look-up" method is effective with benchmark angles
• It relies on memorizing basic trigonometric values

It emphasizes the importance of understanding standard trigonometric values.

Trigonometric Simplification

Trigonometric simplification involves reducing expressions to simpler forms, making complex calculations manageable. Start by evaluating each part of the expression using known identities or values.

For instance, in the problem provided, the expression \( \sqrt{3} \sin \frac{\pi}{6} \) was simplified using the fact that \( \sin \frac{\pi}{6} = \frac{1}{2} \). Thus, the original expression became \( \frac{\sqrt{3}}{2} \). Simplifying trigonometric expressions like this is pivotal; it reduces error chances and eases the subsequent steps of a problem. The technique is fundamental in calculus, physics, and geometry.

• Recognize equivalent forms
• Validate results using well-known ratios

Develop a toolkit of common trigonometric values and identities for faster and more accurate simplifications.