Problem 71 - A Graphical Approach to Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understand the Problem

The problem asks for the exact real value of the expression \( \cos^{-1}(\cos \frac{3\pi}{2}) \). This means we need to find the angle whose cosine is equal to \( \cos \frac{3\pi}{2} \).

Step 2: Determine the Cosine of the Angle

First, compute \( \cos \frac{3\pi}{2} \). The angle \( \frac{3\pi}{2} \) radians corresponds to 270 degrees, and the cosine of 270 degrees is 0. Therefore, \( \cos \frac{3\pi}{2} = 0 \).

Step 3: Apply the Inverse Cosine Function

Next, apply the inverse cosine function \( \cos^{-1}(x) \) to the result from the previous step. \( \cos^{-1}(x) \) is the angle in the interval \([0, \pi]\) whose cosine is \(x\). Since \( \cos^{-1}(0) = \frac{\pi}{2} \), the output of the expression is \( \frac{\pi}{2} \).

Step 4: Confirm the Validity of the Interval

Verify that the solution \( \frac{\pi}{2} \) falls within the range of \( \cos^{-1} \), which is \([0, \pi]\). Since \( \frac{\pi}{2} \) is within this interval, the solution is valid.

Key Concepts

Cosine Function

The cosine function, denoted as \( \cos \), is one of the fundamental trigonometric functions. It relates the angle in a right-angled triangle to the ratio of the adjacent side to the hypotenuse. In the unit circle representation, the cosine of an angle \( \theta \) corresponds to the x-coordinate of the point where the terminal side of the angle intersects the unit circle.
The cosine function is periodic with a period of \( 2\pi \) radians, meaning it repeats its values every \( 2\pi \) interval. This periodicity is crucial when working with angles larger than \( \pi \) radians or \( 180 \) degrees, as in our exercise with \( \frac{3\pi}{2} \).
Cosine values range between \(-1\) and \(1\), with \( \cos 0 = 1 \) and \( \cos \pi = -1 \). Understanding these properties helps in determining the value of the cosine for various angles without a calculator, as certain angles have well-known cosine values.

Radians and Degrees

Radians and degrees are two units of measurement for angles. In trigonometry, the radian is often used because it provides a more natural mathematical language to describe angles. One complete revolution around the circle corresponds to \( 2\pi \) radians or \( 360 \) degrees.
To convert between radians and degrees, you can use the conversions \( 1 \text{ radian} = \frac{180}{\pi} \text{ degrees}\) and \( 1 \text{ degree} = \frac{\pi}{180} \text{ radians} \).
In the original exercise, the angle given is \( \frac{3\pi}{2} \) radians, which equals \( 270 \) degrees. Knowing how to convert and understand these units can assist in visualization and calculation of trigonometric problems.

Trigonometric Angles

Trigonometric angles refer to angles measured in both degrees and radians that correspond to specific points on the unit circle. These angles include all integer multiples of \( \frac{\pi}{2} \) radians (or \( 90 \) degrees), among others. These are often referred to as "special angles" and have known sine and cosine values.
For example,\[\text{Some special trigonometric angles include:}\]

• \(0 \) degrees (\(0 \text{ radians}\)), \( \cos 0 = 1 \)
• \(90 \) degrees (\(\frac{\pi}{2} \text{ radians}\)), \( \cos \frac{\pi}{2} = 0 \)
• \(180 \) degrees (\(\pi \text{ radians}\)), \( \cos \pi = -1 \)
• \(270 \) degrees (\(\frac{3\pi}{2} \text{ radians}\)), \( \cos \frac{3\pi}{2} = 0 \)

Knowing these helps you quickly solve trigonometric problems, like finding the cosine of an angle.

Inverse Functions

Inverse trigonometric functions, such as the inverse cosine function \( \cos^{-1}(x) \), are used to find angles when given trigonometric values. These functions essentially "reverse" the process of their respective trigonometric functions.
• The range of \( \cos^{-1}(x) \) is from \( 0 \) to \( \pi \) radians (inclusive), meaning it will always output an angle within those bounds.
• \( \cos^{-1}(x) \) finds an angle \( \theta \) such that \( \cos \theta = x \).
In the exercise, after converting \( \cos \frac{3\pi}{2} = 0 \), we apply \( \cos^{-1}(0) \), which results in \( \frac{\pi}{2} \).
When dealing with inverse trigonometric functions, remember to verify that the resulting angle makes sense within the function's defined range. This ensures the solution is accurate and meaningful.