Problem 31 - Precalculus with Calculus Previews

Question

Find the exact value of the given trigonometric expression. Do not use a calculator. $$ \cos ^{-1}\left(\cos \left(-\frac{\pi}{4}\right)\right) $$

Step-by-Step Solution

Answer

The exact value is \( \frac{\pi}{4} \).

Step 1: Understand the Inverse Cosine Function

The inverse cosine function, \( \cos^{-1}(x) \), is the angle whose cosine is \( x \). It produces an angle in the range \([0, \pi]\).

Step 2: Apply the Cosine Function

Calculate \( \cos \left(-\frac{\pi}{4}\right) \). Cosine is an even function, so \( \cos(-x) = \cos(x) \). Therefore, \( \cos \left(-\frac{\pi}{4}\right) = \cos \left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \).

Step 3: Find the Inverse Cosine

Use the result from Step 2 to find \( \cos^{-1} \left(\frac{\sqrt{2}}{2}\right) \). The cosine of \( \frac{\pi}{4} \) is \( \frac{\sqrt{2}}{2} \), and \( \cos^{-1} \left(\frac{\sqrt{2}}{2}\right) \) must equal \( \frac{\pi}{4} \) because it's in the range \([0, \pi]\).

Key Concepts

Even and Odd Functions

In trigonometry, understanding the concept of even and odd functions is crucial. These types of functions have specific symmetry properties that simplify calculations.

An even function is symmetric about the y-axis, meaning that applying the function to a negative input gives the same result as with the positive input. In mathematical terms, a function \( f(x) \) is even if \( f(-x) = f(x) \) for all \( x \).

Cosine \( \cos(x) \) is an even function. This means \( \cos(-x) = \cos(x) \), which helps us quickly find \( \cos(-\frac{\pi}{4}) = \cos(\frac{\pi}{4}) \) in our exercise. This property significantly simplifies many trigonometric calculations by negating the need to directly work with negative angles.

Understanding these properties can greatly assist in solving complex problems, as it was strategically used in the solution today.

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for every value in the domain. They are essential tools in simplifying expressions and solving equations.

Some basic trigonometric identities include Pythagorean identities, angle sum and difference identities, and double angle identities. In the presented exercise, we relied on one of these basic properties: the even function property of cosine.

Identifying these identities helps transform complex expressions into simpler ones, making it easier to understand and solve problems.

• The Pythagorean identity for cosine and sine states that \( \cos^2(x) + \sin^2(x) = 1 \).
• Another important identity is \( \tan(x) = \frac{\sin(x)}{\cos(x)} \).
• In practical problems, recognizing these can be the key to arriving at the solution efficiently.

By using such identities, like how \( \cos(-x) = \cos(x) \), the problem is effectively unraveled, leading directly to the correct answer.

Angle Ranges

Angle ranges in trigonometry define the domain in which an angle can be located given a certain function. This is particularly important for inverse trigonometric functions, which restrict standard trigonometric ranges to provide unique solutions.

For example, the inverse cosine function \( \cos^{-1}(x) \) restricts its output to the range \([0, \pi]\). This limited range ensures that each input has exactly one corresponding angle, making inverse functions well-defined for calculations.

In our exercise, this knowledge let us conclude that after applying \( \cos^{-1} \) to \( \frac{\sqrt{2}}{2} \), the angle must be \( \frac{\pi}{4} \), since it falls within the allowed range.

• Other functions have different ranges: \( \sin^{-1}(x) \) gives angles in \([-\frac{\pi}{2}, \frac{\pi}{2}]\).
• Understanding these ranges helps in accurately solving inverse problems.

Thus, mastering the concept of angle ranges is key to becoming proficient with inverse trigonometric functions.