Problem 118 - 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. $$ f(x)=3 \sin x, g(x)=\sin x-1 $$

Step-by-Step Solution

Answer

The intersection points are solutions to the equation 2 sin x + 1 = 0 within the interval [0, 2π]. The solutions should be plugged into any of the original functions to get the y values. Then, these points should be marked on the graph, presenting where f(x) and g(x) intersect.

Step 1: Graph the functions f(x) and g(x) separately

Each function is to be plotted within the given range of 0 ≤ x ≤ 2π. The function f(x) = 3 sin x will have the basic shape of a sine curve but will be vertically stretched by a factor of 3. This means it has an amplitude that is three times taller than the normal sine wave. On the other hand, g(x) = sin x - 1, will look like the standard sine curve, but will be shifted downwards by 1 unit on the y-axis.

Step 2: Plot the functions on the same coordinate system

The two functions should be placed on the same coordinate system to visualize where they intersect. The x-axis will represent x values within the interval [0, 2π] and the y-axis will represent the values of f(x) and g(x).

Step 3: Find intersection points

To find the intersection points, set the two functions equal to each other and solve for x. Do it by solving 3 sin x = sin x - 1. This will result in an equation 2 sin x + 1 = 0. Solving this equation will give the x-values where f(x) and g(x) intersect.

Step 4: Identify intersection points on the graph

Plug the derived x-values into any of the original functions to get the corresponding y-values. These (x,y) coordinates are the intersection points of f(x) and g(x), which should also be identifiable on the plot.

Key Concepts

Sine Function

The sine function is a foundational trigonometric function that repeats its pattern in predictable intervals, known as periods. It is part of the family of functions that are collectively used to model periodic phenomena. Here’s what you need to know about the sine function:

• Basic Form: The sine function is generally written as \( \sin(x) \), where \( x \) is the angle measured in radians.
• Amplitude: For \( f(x) = \sin(x) \), the maximum height, or amplitude, is 1, and similarly, the minimum is -1.
• Periodicity: The sine wave repeats every \(2\pi\) radians.

For the function \( f(x) = 3 \sin(x) \), the amplitude is multiplied by 3, stretching the wave vertically. This increases the maximum peak to 3 and the minimum valley to -3, while the period remains \(2\pi\). Understanding these properties helps us graph and analyze sine functions effectively.

Intersection Points

Intersection points occur where two graphs, such as \( f(x) \) and \( g(x) \), meet or cross each other. Discovering these points involves solving equations where the functions equal each other.

• For functions \( f(x) = 3 \sin(x) \) and \( g(x) = \sin(x) - 1 \), set them equal to find where they intersect: \(3 \sin(x) = \sin(x) - 1\).
• Simplify the equation to find intersection points. Rearranging gives \(2 \sin(x) + 1 = 0\).

Solving for \( x \), we have \( \sin(x) = -\frac{1}{2} \). This trigonometric equation yields solutions at specific angles within the interval \(0 \leq x \leq 2\pi\), indicating the intersections you can mark on the graph.

Graphing Functions

Graphing trigonometric functions involves plotting their points on a coordinate plane, reflecting their period, amplitude, and any shifts. For example:

• Function \( f(x) = 3 \sin(x) \): This graph reflects a sine function with an amplitude of 3. It oscillates between 3 and -3, over a period of \(2\pi\).
• Function \( g(x) = \sin(x) - 1\): Here, the graph represents a normal sine wave, shifted downward by 1 unit on the y-axis.

By placing these graphs on the same coordinate plane, any intersection points become visually evident. The overlap or crossing points signify where the solutions to their equations line up. Understanding how to properly graph these types of functions aids in visually solving and confirming mathematical problems.

Trigonometric Equations

Trigonometric equations involve finding angles that satisfy equations involving trigonometric functions. For our functions:

• Our primary task was solving \( 2 \sin(x) + 1 = 0 \), derived from equating \( 3 \sin(x) \) and \( \sin(x) - 1 \).
• This simplifies to solving \( \sin(x) = -\frac{1}{2} \), a standard sinusoidal problem.
• Utilize known values: sine achieves a negative half at specific angles within the unit circle, like \(x = \frac{7\pi}{6} \) and \(x = \frac{11\pi}{6} \) within \(0\) to \(2\pi\).

These solutions are then used to determine where the graphs of the functions intersect. Mastery of solving such trigonometric equations is crucial for effectively managing and interpreting the broader use of trigonometric functions in various applications.