Problem 99 - Algebra and Trigonometry

Question

Graph each equation in a \(\left[-2 \pi, 2 \pi, \frac{\pi}{2}\right] b y\) \([-3,3,1]\) 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 \( y=\frac{1-2 \cos 2x}{2\sin x-1} \) can also be described by the equation \( y= - \cot x - \cos x \). It has been verified that both equations are equivalent.

Step 1: Plot the graph

The equation \( y=\frac{1-2 \cos 2x}{2\sin x-1} \) is plotted within the viewing rectangle [-2 pi, 2 pi, pi/2] by [-3,3,1].

Step 2: Analyze the graph and derive a new equation

Upon observing, the nature of the graph leads to the formulation of a new equation that matches the original one. After performing several manipulations and substitutions using trigonometric identities, the equation that can describe the same graph is derived as \( y= -\cot x - \cos x \)

Step 3: Verify both equations are equivalent

To prove that the two equations are equivalent, write \( y=\frac{1-2 \cos 2x}{2\sin x-1} \) in a new form using trigonometric identities and rewriting techniques. After a series of manipulations, the equation becomes \( y= - \cot x - \cos x \), proving that both equations indeed describe the same graph.

Key Concepts

Graphing Trigonometric Functions

When addressing trigonometric equations, graphing can give us a fantastic visual perspective. By visualizing the graph of an equation like \( y=\frac{1-2 \cos 2x}{2\sin x-1} \), we can discern key characteristics such as symmetry, periodicity, and specific behavior changes.
In this exercise, the graph is plotted over the interval \([-2\pi, 2\pi]\), with a viewing window ranging from -3 to 3 on the y-axis. This range captures several cycles of typical trigonometric patterns and helps spot periodic behavior.

Graphing is useful because:

• You can visually compare graphs to identify equivalent equations.
• It highlights critical points like intersections, maxima, and minima.
• Detects asymptotic behavior where functions approach but don't touch certain values.

Once a graph is plotted, slight variations in equation form but identical graphs suggest equivalent equations using different trigonometric identities.

Trigonometric Identities

Trigonometric identities are essential tools to simplify or transform trigonometric expressions. They allow mathematicians to express trigonometric functions in different forms, which can facilitate deeper understanding and simpler computations.
In this exercise, identities play a critical role in simplifying \( y=\frac{1-2 \cos 2x}{2\sin x-1} \) into a more manageable form. The derived equivalent form, \( y = -\cot x - \cos x \), showcases how identities can redefine expressions while preserving equivalence.

Some common trigonometric identities useful in such transformations include:

• Pythagorean identities like \( \sin^2x + \cos^2x = 1 \).
• Double-angle identities such as \( \cos 2x = 2\cos^2x - 1 \).
• Reciprocal identities, for example, \( \cot x = \frac{1}{\tan x} \).

By selecting appropriate identities, students can simplify complex trigonometric equations and discover new equivalent expressions that may be easier to work with or graph.

Equivalent Equations

Equivalent equations are different expressions that yield the same results for all values within their domain. In the context of trigonometric equations, using identities to transform an equation can reveal an equivalent form that might be simpler to understand or solve.
For this exercise, showing that \( y=\frac{1-2 \cos 2x}{2\sin x-1} \) is equivalent to \( y= - \cot x - \cos x \) can be verified through algebraic manipulations and substituting identities.

Key steps to verify equations are equivalent include:

• Using identities to rewrite one equation in the form of another.
• Performing algebraic manipulations, such as factoring or expanding.
• Substituting common values into both equations to ensure they yield the same results.

Successfully demonstrating equivalency requires thoughtful application of trigonometric identities and can deepen understanding of mathematical relationships across different forms of the same equation. This skill is not only pivotal in solving academic problems but also in practical applications, where multiple forms of an equation might be useful for different purposes.