Problem 99 - Precalculus Essentials

Question

In Exercises \(99-101,\) graph each equation in \(a\left[-2 \pi, 2 \pi, \frac{\pi}{2}\right]\) by [-3,3,1] viewing rectangle. Then a. Describe the graph using another equation, and b. Verify that the two equations are equivalent. $$y=\frac{1-2 \cos 2 x}{2 \sin x-1}$$

Step-by-Step Solution

Answer

The graph of the equation \(y = \frac{1-2\cos 2x}{2\sin x - 1}\) appears similar to the function \(y = \tan x\). After verification using a set of \(x\) values to check both equations, it's accurate to say that the first equation is an equivalent form of \(y = \tan x\) within the specified range.

Step 1: Graph the Given Equation

Make a table for the values of \(x\) from \(-2\pi\) to \(2\pi\) with a step of \(\pi/2\) and calculate the respective \(y\) values using the equation \(y = \frac{1-2\cos 2x}{2\sin x - 1}\). Use these values to plot the function on a graph.

Step 2: Describe the Graph with Another Equation

Observe the graph and its characteristics: its period, its amplitude, its maxima and minima, and its phase shift. With these observations, express the graph with another equivalent trigonometric equation. Here let's take \(y = \tan x\).

Step 3: Verification

To verify that the two equations are equivalent, substitute the same set of \(x\) values into both equations. If both equations yield the same \(y\) values within an acceptable error range, then the two equations can be considered equivalent.

Key Concepts

Trigonometric Equations

Trigonometric equations are mathematical expressions involving trigonometric functions that can be used to describe various periodic phenomena, from the motion of waves to the oscillation of a pendulum. Solving these equations requires an understanding of trigonometric identities and how they transform. For instance, the given exercise presents the equation
\[ y=\frac{1-2 \cos 2x}{2\sin x-1} \]
To graph this equation, one would typically isolate the trigonometric functions and look at their respective periods and transformations to outline the basic shape. However, the process might involve additional steps like factoring and using identities to simplify the expression.

Amplitude and Period of Trig Functions

The amplitude of a trigonometric function refers to the height of the wave, or the maximum value it reaches from its central axis. The period refers to the length of one complete cycle of the wave. For example, in a standard cosine function,
\[ y = A \cos(Bx+C) \]
the amplitude is represented by the absolute value of \(A\) and the period is calculated by \(\frac{2\pi}{|B|}\). When graphing trig functions like
\[ y = (1-2 \cos 2x) \]
from the given exercise, it’s important to first identify these parameters. The coefficient 2 in front of \(\cos\) affects the period of the cosine function, halving it from the usual \(2\pi\) to \(\pi\). Recognizing these characteristics is crucial for accurately plotting the graph and understanding its behavior within the specified interval.

Phase Shift of Trigonometric Graphs

Phase shift in trigonometric graphs describes the horizontal displacement from the usual position of the trigonometric function's wave. If we consider a trigonometric equation of the form
\[ y = A \cos(Bx+C) + D\]
or
\[ y = A \sin(Bx+C) + D \]
then the phase shift is given by \(-\frac{C}{B}\). A positive phase shift means the graph is shifted to the left, while a negative phase shift moves it to the right. In the exercise example, the \(\cos\) function does not have an explicit phase shift, whereas in another form, such as
\[ y = \tan(x) \]
used to describe the same graph, the function inherently starts at a different position than a regular tangent function because it has been transformed through other trigonometric operations. Understanding phase shifts is vital for aligning the graph correctly and interpreting or predicting the nature of the wave’s motion over its period.