Problem 32 - Precalculus Essentials

Question

Find the exact value of each expression, if possible. Do not use a calculator. $$\cos \left(\cos ^{-1} 0.57\right)$$

Step-by-Step Solution

Answer

The exact value of the given expression is 0.57.

Step 1: Understanding the Problem

We are given the expression \(\cos(\cos^{-1} 0.57)\). The task is to find the exact value of the entire expression, not using any calculator. We are basically required to find the cosine of the angle whose cosine is 0.57.

Step 2: Applying Function to its Inverse

We know that an inverse function 'undoes' the operation of the original function. So when computed together, we get the original value back. Hence, \(\cos(\cos^{-1} 0.57) = 0.57\).

Key Concepts

Cosine Function

The cosine function is one of the basic functions encountered in trigonometry. It's often represented as \(\cos(\theta)\), where \(\theta\) is the angle in a right-angled triangle opposite the adjacent side to \(\theta\), while the hypotenuse is the triangle's longest side.

The cosine function maps the ratio of the lengths of the adjacent side to the hypotenuse of a right-angled triangle for a given angle. In the unit circle context, it represents the x-coordinate of a point where the line at an angle \(\theta\) intersects the circle. In mathematical terms, \(\cos(\theta) = \frac{\text{adjacent side}}{\text{hypotenuse}}\).

In the context of the given problem, \(\cos(\cos^{-1} 0.57)\) denotes the cosine of an angle which is itself the result of an inverse cosine operation. This can be a bit mind-bending, but it simply means we're looking for the value associated with an angle whose cosine is 0.57. This reveals an important property: applying \(\cos\) to its inverse \(\cos^{-1}\) (and vice versa) returns the initial value. Therefore, \(\cos(\cos^{-1} 0.57)=0.57\) because they are 'function inverses' of each other.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables involved. These identities are useful for simplifying complex expressions, solving equations, and transforming the form of a trigonometric expression.

Some of the fundamental trigonometric identities include the Pythagorean identities (e.g., \(\cos^2(\theta) + \sin^2(\theta) = 1\)), reciprocal identities (e.g., \(\sec(\theta) = \frac{1}{\cos(\theta)}\)), and angle sum or difference identities (e.g., \(\cos(\alpha + \beta) = \cos(\alpha)\cos(\beta) - \sin(\alpha)\sin(\beta)\)).

In our exercise, we use the identity that relates the function to its inverse to simplify the expression: \(\cos(\cos^{-1}(x)) = x\), given that -1≤x≤1. This identity is an application of a broader principle where any function and its inverse will cancel each other out, returning the original input, given that the input is in the domain of the trigonometric function.

Exact Values of Trigonometric Functions

An integral part of trigonometry is knowing the exact values of trigonometric functions for commonly used angles. These are angles like 0°, 30°, 45°, 60°, and 90°, among others. For example, \(\cos(0°) = 1\), \(\cos(60°) = \frac{1}{2}\), and \(\cos(90°) = 0\).

For these specific angles, one can memorize their sine, cosine, and tangent values. However, for angles that do not have such well-known exact values, it is important to understand how to find or approximate these values when necessary. In the context of our exercise, \(\cos^{-1}(0.57)\) might not be one of those angles with a memorized exact value, but since we're applying the cosine function to its inverse, we can find the exact value without needing to know the angle itself. The exercise leverages the concept of inverse functions to find the cosine of an angle without actually determining the angle. This simplifies the process significantly and showcases the deep relationships within trigonometry.