Problem 69 - Precalculus: A Concise Course

Question

Use a graphing utility to graph the two equations in the same viewing window. Use the graphs to determine whether the expressions are equivalent. Verify the results algebraically. $$ y_{1}=\frac{\cos x}{\sin x}, \quad y_{2}=\cot x $$

Step-by-Step Solution

Answer

The expressions \(y_{1}=\frac{\cos x}{\sin x}\) and \(y_{2}=\cot x\) are indeed equivalent. This was verified both graphically and algebraically.

Step 1: Graphing the equations

Use a graphing utility to plot the graphs of both \(y_{1}=\frac{\cos x}{\sin x}\) and \(y_{2}=\cot x\). Analyze the patterns, shapes, and intersections.

Step 2: Comparing the graphs

Look at the graphs of both \(y_{1}\) and \(y_{2}\). Do they overlap completely? If yes, they are likely to be equivalent. If there are parts where they do not overlap, note those down.

Step 3: Algebraic verification

The two functions can be equated: \[\frac{\cos x}{\sin x} = \cot x\]Now, it is known from the definitions of trigonometric functions that \(\cot x\) is actually \(\frac{\cos x}{\sin x}\). Hence, both the expressions are indeed equivalent.

Key Concepts

Graphing utility

Graphing utilities are powerful tools that help visualize mathematical functions, including trigonometric identities. These tools are typically found in graphing calculators or software programs. By entering different equations, users can observe their graphical representation.

For the given exercise, you need to use a graphing utility to plot the two equations \(y_{1}=\frac{\cos x}{\sin x}\) and \(y_{2}=\cot x\). Using a graphing utility, these functions can be visualized together in the same viewing window. This allows for a comparison to determine if the two equations represent the same curve. Simply enter each equation into the utility and choose an appropriate window or scale for a clear view.

When viewing these graphs, look for:

• Their paths: Do they share the same path or trajectory?
• Overlapping: Do both graphs align perfectly throughout the domain?

These observations suggest whether the two functions might be equivalent. Such graphical comparisons can often reveal equivalence before delving into algebraic verification.

Algebraic verification

Algebraic verification is the process of using algebra to prove or disconfirm the equivalence between two expressions. In this exercise, after visually suggesting equivalency through graphs, the next step is algebraic verification.

Let's consider the expressions given: \(y_{1}=\frac{\cos x}{\sin x}\) and \(y_{2}=\cot x\). To verify algebraically, recall the definition of the cotangent function: \(\cot x = \frac{\cos x}{\sin x}\).

This directly matches the form of \(y_{1}\), confirming their equivalency mathematically. Hence, no further simplification is required. This straightforward comparison is often the case with trigonometric identities, particularly when definitions directly support the equivalence.

Trigonometric functions

Trigonometric functions are fundamental in mathematics, often used to describe the relationships within triangles, especially right-angled ones. Important trigonometric functions include sine ( ext{sin}), cosine ( ext{cos}), tangent ( ext{tan}), and their reciprocals, such as cotangent ( ext{cot}), secant ( ext{sec}), and cosecant ( ext{csc}).

In this context, \(y_{1} = \frac{\cos x}{\sin x}\) utilizes cosine and sine to define a ratio, known as the cotangent ( ext{cot}). This relationship emerges from the basic identities:

• \(\cot x = \frac{1}{\tan x} = \frac{\cos x}{\sin x}\)
• \(\tan x = \frac{\sin x}{\cos x}\)

By knowing these fundamental identities, many trigonometric expressions can be rewritten or identified.

Understanding these functions and their relational interchanges helps when dealing with more complex problems or verifying expressions, revealing deeper insights into trigonometric identities.