Problem 157 - Precalculus

Question

For what value of \(x\) does \(\cos x=\cos ^{-1} x ?\) Use a graphing calculator to approximate the answer.

Step-by-Step Solution

Answer

The approximate value of \(x\) is 0.739.

Step 1: Understanding the Problem

We need to find the value of \(x\) for which \(\cos x = \cos^{-1} x\). \(\cos x\) is the cosine function, and \(\cos^{-1} x\) (also written as \(\arccos x\)) is the inverse cosine function.

Step 2: Analyzing Domain and Range

The range of \(\cos^{-1} x\) is \([0, \pi]\) and the domain is \([-1, 1]\). Consequently, we need to look for \(x\) within this interval such that \(\cos x\) and \(\cos^{-1} x\) are equal.

Step 3: Graphical Approach

Plot the graphs of \(y = \cos x\) and \(y = \cos^{-1} x\) over the interval \([0, \pi]\). The solution to \(\cos x = \cos^{-1} x\) will be the point(s) of intersection of these two curves.

Step 4: Using the Graphing Calculator

Use a graphing calculator to find the intersection point(s) of \(y = \cos x\) and \(y = \cos^{-1} x\). Use the intersection feature of the calculator over the interval \([0, \pi]\).

Step 5: Identifying the Intersection

From the graph, identify the point of intersection, which is approximately \(x \approx 0.739\). This is the value of \(x\) that satisfies the equation \(\cos x = \cos^{-1} x\).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are fascinating tools in mathematics. They allow us to find angles when given a trigonometric ratio. Specifically, the \ \( \cos^{-1} x \ \) function, also known as the inverse cosine or arccosine, is used to determine the angle whose cosine is \ \( x \ \). Unlike regular cosine, which maps angles to values between -1 and 1, the inverse cosine maps within the range \ \([0, \pi]\). - **Key properties**:
- **Domain**: \ \([-1, 1]\)
- **Range**: \ \([0, \pi]\)
It's important to understand that inverse trigonometric functions work with angles in radians by default. This knowledge is crucial when solving problems involving equalities like \ \( \cos x = \cos^{-1} x \ \). Understanding the relationship between \ \( x \ \) and \ \( \cos^{-1} x \ \) helps us identify solutions within specified intervals.

Graphing Calculator

A graphing calculator is indispensable when dealing with complex equations involving trigonometric functions. They provide an intuitive way to visualize functions and understand their interactions. For the problem at hand, we used a graphing calculator to find where \ \( y = \cos x \ \) and \ \( y = \cos^{-1} x \ \) intersect. - **Benefits of using a graphing calculator**:
- Provides a visual representation of equations
- Allows easy manipulation of graphs
- Finds intersection points of functions in a given rangeThe graphing calculator simplifies the task of approximating values of \ \( x \ \) where these functions intersect by directly using the intersection feature. This feature is especially useful in narrowing down specific solutions over restricted domains.

Cosine Function

The cosine function, represented as \ \( \cos x \ \), is a fundamental trigonometric function that outputs the cosine of an angle \ \( x \ \). It is periodic with a cycle of \ \( 2\pi \ \), meaning it repeats its wave-like pattern every \ \( 2\pi \ \) units.- **Properties of \ \( \cos x \ \):**
- **Range**: \ \([-1, 1]\)
- **Period**: \ \(2\pi\)
- Symmetrical about the y-axis (even function)Understanding the cosine function is essential when equating it to its inverse, as it helps in identifying the intervals where both functions might intersect. The symmetry and periodic nature of \ \( \cos x \ \), combined with its bounded range, make it critical to analyze it properly to solve equations like \ \( \cos x = \cos^{-1} x \ \).

Intersection Points

Finding intersection points between two functions involves identifying where their graphs converge at the same value of \ \( x \ \). In our exercise, this concept is applied by checking where \ \( y = \cos x \ \) meets \ \( y = \cos^{-1} x \ \). Through graphical methods, like using a graphing calculator, we identify these intersection points accurately.- **Why intersection points matter**:
- They represent solutions to where two equations are equal
- Provide an effective way to understand function interactions
In practical terms, for \ \( \cos x = \cos^{-1} x \ \), the intersection point within the interval \ \([0, \pi]\) is approximately \ \( x \approx 0.739 \ \). Identifying such points graphically aids in grasping how trigonometric functions behave relative to each other within their domains.