Problem 120 - Algebra and Trigonometry

Question

Graph \(f\) and \(g\) in the same rectangular coordinate system for \(0 \leq x \leq 2 \pi .\) Then solve a trigonometric equation to determine points of intersection and identify these points on your graphs. $$ f(x)=\cos 2 x, g(x)=1-\sin x $$

Step-by-Step Solution

Answer

The functions \(f(x) = \cos(2x)\) and \(g(x) = 1 - \sin(x)\) intersect at points where the trigonometric equation \(\cos(2x) = 1 - \sin(x)\) holds true. To find the exact points, this equation needs to be solved and the solutions should then be plotted onto the graphs.

Step 1: Graph the functions

Each function should be plotted on a graph for the domain \(0 \leq x \leq 2\pi\). Plot \(f(x) = \cos(2x)\) and \(g(x) = 1 - \sin(x)\) separately using a graphing tool or graphing calculator.

Step 2: Combine the graphs

To visualize the functions in the same rectangular coordinate system, superimpose the graphs of \(f(x)\) and \(g(x)\) on the same set of axes.

Step 3: Find the points of intersection

Set the two functions equal to each other and solve the resulting equation, i.e., \(\cos(2x) = 1 - \sin(x)\), for \(x\). This requires conversion of the equation into a format that can be solved using common trigonometric identities. The solutions to this equation represent the x-coordinates of the points of intersection. Then, substitute these x-values into either of the original functions to obtain the y-coordinates.

Step 4: Identify the points on the graphs

Mark the points of intersection on the graph. The coordinates of the points of intersection are those calculated in step 3.

Key Concepts

Function Intersection

When exploring trigonometric graphs, identifying where two functions intersect is an essential skill. Intersection of two functions occurs where their graphs touch or cross each other. To find these points, you must solve the equation obtained by setting the functions equal to each other.
For this exercise, where the functions are \( f(x) = \cos(2x) \) and \( g(x) = 1 - \sin(x) \), you set \( \cos(2x) = 1 - \sin(x) \).
Solving this requires a deep understanding of trigonometric identities and techniques like rearranging terms or using substitution methods.
The solutions to this equation give the x-coordinates where the intersections occur. Once these x-values are found, substituting back into any of the initial functions will provide the corresponding y-values of these points.

Graphing Trigonometric Functions

Understanding how to graph trigonometric functions like \( \cos(2x) \) and \( 1 - \sin(x) \) is crucial for this exercise. Each trigonometric function has unique properties such as amplitude, period, phase shift, and vertical shift.

• The function \( f(x) = \cos(2x) \) is a cosine wave with a period that is half that of the normal cosine function, because the factor of 2 inside the cosine affects the frequency.
• On the other hand, \( g(x) = 1 - \sin(x) \) involves a vertical shift due to the constant 1 subtracted from the sine function.

When graphing, it's important to note these shifts and compressions or stretches. Begin by plotting each function individually over the domain \( 0 \leq x \leq 2\pi \). Then, superimpose them on one set of axes to observe the complete picture, facilitating the identification of intersection points visually.

Trigonometric Identities

Trigonometric identities are the cornerstone for solving trigonometric equations. In the context of finding function intersections, identities can simplify expressions and make solving equations feasible.
Some important identities that might be useful are:

• Double angle identities, such as \( \cos(2x) = \cos^2(x) - \sin^2(x) \).
• Sum and difference identities, which can rewrite expressions in terms of sine and cosine.

In the equation \( \cos(2x) = 1 - \sin(x) \), recognizing these identities and knowing how to apply them can transform the equation into a more solvable form.
These tools allow you to unravel complex equations by breaking them down into familiar patterns, helping to pinpoint where graphs will intersect.