Problem 17 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The exact value is \( \frac{\pi}{3} \).

Step 1: Identify the Inner Expression

The problem asks for the exact value of \( \cos^{-1}\left(\cos \frac{\pi}{3}\right) \). The inner expression here is \( \cos \frac{\pi}{3} \).

Step 2: Simplify the Inner Expression

Calculate \( \cos \frac{\pi}{3} \). Knowing that \( \frac{\pi}{3} \) is a standard angle on the unit circle, we find that \( \cos \frac{\pi}{3} = \frac{1}{2} \).

Step 3: Evaluate the Inverse Function

Now evaluate \( \cos^{-1}\left(\frac{1}{2}\right) \). We know that the range of \( \cos^{-1}(x) \) is \( [0, \pi] \), and within this range, the angle whose cosine is \( \frac{1}{2} \) is \( \frac{\pi}{3} \).

Step 4: Verify the Conditions

Ensure that the angle \( \frac{\pi}{3} \) is indeed within the range of the inverse cosine function, which is from 0 to \( \pi \). Since \( \frac{\pi}{3} \approx 1.047 \), it lies within this range.

Key Concepts

Cosine Function

The cosine function, often represented as \( \cos(\theta) \), is a fundamental part of trigonometry. It relates the angle \( \theta \) to the ratio of the adjacent side over the hypotenuse in a right triangle. This definition forms the basis for its representation on the unit circle.

• The function is periodic with a period of \( 2\pi \), meaning that it repeats its values every \( 2\pi \).
• The range of the cosine function is between -1 and 1, because it represents a ratio of lengths.

Understanding the cosine function is crucial for solving trigonometric problems, especially when dealing with angles and cycles of waves. Calculating values of the cosine function for standard angles is often the first step in solving more complex trigonometric expressions.

Unit Circle

The unit circle is a circle centered at the origin of a coordinate plane with a radius of one. It is a powerful tool for understanding trigonometric functions and their values.

• Each point on the unit circle can be represented as \((\cos(\theta), \sin(\theta))\), where \(\theta\) is the angle formed with the positive x-axis.
• It allows us to visualize how the cosine and sine values change with different angles.

For the angle \( \frac{\pi}{3} \) on the unit circle, the coordinates are \((\frac{1}{2}, \frac{\sqrt{3}}{2})\). This immediately shows that \( \cos(\frac{\pi}{3}) = \frac{1}{2} \). Utilizing the unit circle aids greatly in visualizing and solving trigonometric expressions by illustrating these relationships clearly.

Standard Angles

In trigonometry, certain angles are referred to as "standard angles." These are angles typically found in right triangle trigonometry and the unit circle:

• \(0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2} \)
• Their respective cosine values are \(1, \frac{\sqrt{3}}{2}, \frac{\sqrt{2}}{2}, \frac{1}{2}, 0\)

Standard angles simplify computations because their trigonometric values are well known and memorized. They are especially helpful when working with inverse trigonometric functions like \( \cos^{-1} \), as these standard values often appear in solutions. In the example with \( \cos^{-1}\left(\cos \frac{\pi}{3}\right) \), recognizing \( \frac{\pi}{3} \) as a standard angle allows us to directly find the inverse cosine value.

Trigonometric Identities

Trigonometric identities are mathematical equations that involve trigonometric functions and are true for every value of the occurring variables. They are essential tools that simplify solving complex trigonometric problems.

• One key identity is the Pythagorean identity: \( \cos^2(\theta) + \sin^2(\theta) = 1 \).
• Another is the arc cosine identity, where \( \cos(\cos^{-1}(x)) = x \) for \( x \) in the range [-1, 1].

In the context of solving \( \cos^{-1}\left(\cos \frac{\pi}{3}\right) \), knowing that the inverse cosine function cancels out with cosine reveals redundancy when the angle is within the principal range of \([0, \pi]\). Trigonometric identities provide shortcuts and validate the steps when solving otherwise complex trigonometric equations.