Problem 48 - Algebra and Trigonometry Real Mathematics, Real People

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

Yes, the given expressions \(y_{1}=\frac{\cos x}{\sin x}\) and \(y_{2}=\cot x\) are equivalent as determined visually from the graph and confirmed algebraically.

Step 1: Graphing Expressions

Using a graphing tool, input both expressions \(y_{1}=\frac{\cos x}{\sin x}\) and \(y_{2}=\cot x\). Ensure that they are both plotted on the same viewing window for easy comparison.

Step 2: Visual comparison

Observe the two graphs. If they appear to be the same, then it is likely that the expressions are equivalent. However, this visual comparison is not enough for a thorough understanding, thus algebraic confirmation is needed.

Step 3: Algebraic confirmation

The algebraic confirmation involves simplifying one or both of the expressions until they look the same. By definition, \(\cot x = \frac{\cos x}{\sin x}\). So, algebraically, these two expressions are equivalent.

Key Concepts

Graphing Utilities for Trigonometric Functions

Using graphing utilities or graphing calculators is a fantastic way to visualize mathematical functions and expressions. In this exercise, we have two trigonometric expressions:

• \( y_1 = \frac{\cos x}{\sin x} \)
• \( y_2 = \cot x \)

To explore these, input both expressions into a graphing utility like Desmos or a scientific calculator. Ensure that these expressions are plotted in the same viewing window. This allows you to directly compare their shapes and behavior over the same interval of x-values.
Graphing utilities help in quickly identifying patterns and similarities between different expressions. While observing, ensure both graphs extend over several cycles of the trigonometric function to see if their periodicity aligns.
Look for overlapping or matching points, shapes, and continuity. Such observation establishes an initial inference regarding their equivalence, though further confirmation is required.

Visual Comparison of Graphs

The essence of visual comparison lies in examining the two graphs plotted on the same axis to see if they follow the same path. Here, when you view the graphs of \( y_1 = \frac{\cos x}{\sin x} \) and \( y_2 = \cot x \), look for:

• Coinciding lines or curves
• Identical peaks, troughs, and points of intersection
• Similar periodic behavior

If both graphs overlap completely across several cycles, it suggests strongly that the expressions might be equivalent. Visual comparison is quick and helps develop an intuitive sense of equivalence.
However, it's important to remember that a visual inspection is a preliminary step. Graphs may look similar visually, but due to resolution or axis limits, small differences might not be evident. Thus, to ensure absolute precision, additional steps are required.

Algebraic Confirmation of Expression Equivalence

Algebraic confirmation complements visual comparison by analytically proving that two expressions are equivalent. After identifying the potential equivalence visually between \( y_1 = \frac{\cos x}{\sin x} \) and \( y_2 = \cot x \), the next step is demonstrating this through algebra.
Recall the identity:

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

This identity directly shows that \( y_1 \) and \( y_2 \) are equivalent since \( \cot x \), by trigonometric definition, is \( \frac{\cos x}{\sin x} \). This algebraic step is crucial as it provides a definitive answer beyond the approximation or potential errors of graphing tools.
It involves understanding and applying known trigonometric identities to transform one expression into another, showing they are simply different forms of the same relationship.

Understanding Equivalent Expressions

Equivalent expressions are expressions that result in the same value for any value of the variable within their domain. For \( y_1 = \frac{\cos x}{\sin x} \) and \( y_2 = \cot x \), both expressions simplify to the same fundamental trigonometric identity, illustrating their equivalency.
When dealing with equivalent expressions:

• They look different but simplify to the same form.
• For trigonometric functions, identities can help recognize equivalency.
• They must hold true for all values in their domain—not just at specific points.

In algebra, discovering equivalent expressions often involves factoring, expanding, combining, or using identities. Grasping equivalence enriches understanding of mathematical relationships and interconnections among various forms of equations and expressions.