Problem 68 - A Graphical Approach to Precalculus with Limits

Question

Give the exact real number value of each expression. Do not use a calculator. $$\cos \left(\cos ^{-1} \frac{\sqrt{3}}{2}\right)$$

Step-by-Step Solution

Answer

The value is \( \frac{\sqrt{3}}{2} \).

Step 1: Understanding the Inverse Function

We begin by interpreting the expression. Here, \( \cos^{-1} \left(\frac{\sqrt{3}}{2}\right) \) represents the angle whose cosine is \( \frac{\sqrt{3}}{2} \). This means we're finding the angle such that when we take the cosine of this angle, we retrieve \( \frac{\sqrt{3}}{2} \).

Step 2: Recognizing Known Values

Recognize that \( \frac{\sqrt{3}}{2} \) is a special value associated with a common angle in the unit circle. We know that the cosine of \( \frac{\pi}{6} \) (or 30 degrees) is equal to \( \frac{\sqrt{3}}{2} \). Therefore, \( \cos^{-1} \left( \frac{\sqrt{3}}{2} \right) = \frac{\pi}{6} \).

Step 3: Evaluating the Cosine of the Angle

Now we take the cosine of the angle we found. So, \( \cos \left( \frac{\pi}{6} \right) \) needs to be evaluated. We know from the unit circle that \( \cos \left( \frac{\pi}{6} \right) = \frac{\sqrt{3}}{2} \).

Step 4: Providing the Answer

Since we evaluated \( \cos \left( \cos^{-1} \left( \frac{\sqrt{3}}{2} \right) \right) \) to be \( \frac{\sqrt{3}}{2} \), we conclude that the value of the given expression is \( \frac{\sqrt{3}}{2} \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are crucial in solving various trigonometric equations, especially when determining an angle from a function value. They essentially serve as the opposite operation to normal trigonometric functions, allowing us to work backwards to find angles. For example, the inverse cosine function, written as \( \cos^{-1}(x) \), is the angle \( \theta \) for which \( \cos(\theta) = x \).
When we say \( \cos^{-1} \left(\frac{\sqrt{3}}{2} \right) \), we're searching for the angle whose cosine is \( \frac{\sqrt{3}}{2} \).
This process is like asking, "What angle, if I input into the cosine function, will give me back \( \frac{\sqrt{3}}{2} \)?". In the context of common angles, this scenario often appears in exercises aimed at mastering the function's logic and various functionalities.
Understanding these inverse functions helps simplify complex expressions and solve trigonometric problems that provide insights into periodic phenomena like waves or circular motion.

Unit Circle

The unit circle is an essential concept in understanding trigonometric functions, particularly when visualizing and solving problems involving them. It is a circle with a radius of 1, centered at the origin of a coordinate plane, and is used to define sine, cosine, and tangent functions for all angles.
Each point on the unit circle can be expressed in terms of \((x, y)\) coordinates, where \(x\) corresponds to the cosine value, and \(y\) represents the sine value of a given angle. This makes it simple to calculate trigonometric values for common angles, like \(\frac{\pi}{6}\), \(\frac{\pi}{4}\), and \(\frac{\pi}{3}\).
In our exercise, knowing that \( \cos \left( \frac{\pi}{6} \right) = \frac{\sqrt{3}}{2} \) is straightforward with the unit circle. This familiarity makes it easy to identify and recall special angles associated with particular trigonometric values.
One of the most powerful aspects of using the unit circle is how it helps us interpret and predict the behavior of trigonometric functions across different quadrants, making it a foundational tool for anyone studying trigonometry.

Special Angles

Special angles are specific angles that have well-known trigonometric values, which makes them highly important in mathematical calculations. These angles usually refer to \(0\), \(\frac{\pi}{6}\), \(\frac{\pi}{4}\), \(\frac{\pi}{3}\), \(\frac{\pi}{2}\), and multiples thereof. Each of these angles corresponds to easily remembered trigonometric values.
In the context of inverse trigonometric functions, special angles help us quickly find solutions. For instance, knowing that \( \cos \left( \frac{\pi}{6} \right) = \frac{\sqrt{3}}{2} \) means we can immediately determine that \( \cos^{-1} \left( \frac{\sqrt{3}}{2} \right) = \frac{\pi}{6} \).
These angles are particularly useful because they appear often in real-world contexts and standardized tests, which expect students to recognize and use these values fluently.
Mastering these angles helps students improve their understanding of trigonometry and enables them to solve problems more efficiently, forming a base on which more advanced mathematical concepts are built.