Problem 68 - 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. $$ y_{1}=\sin x \sec x, \quad y_{2}=\tan x $$

Step-by-Step Solution

Answer

By plotting \(y_{1}=\sin x \sec x\) and \(y_{2}=\tan x\) in the same graphing window and comparing the graphs, it can be concluded that these expressions are equivalent. This is also algebraically confirmed using trigonometric identities, as \(\sin x \sec x\) simplifies to \(\tan x\).

Step 1: Plotting the Graphs

Graph the functions \(y_{1}=\sin x \sec x\) and \(y_{2}=\tan x\) on the same graphing utility. Make sure to use a common scale for the x and y axes.

Step 2: Visual Comparisons of Graphs

Study the graphs closely. If the graphs coincide completely, it means the two equations represent the same function, i.e., the expressions are equivalent.

Step 3: Algebraic Verification

According to the trigonometric identities, \(\sec x\) is equivalent to \(1/\cos x\), so we can rewrite \(\sin x \sec x\) as \(\sin x \times 1/\cos x\) which simplifies to \(\tan x\). This rework confirms the fact that \(y_{1}\) and \(y_{2}\) are equivalent expressions.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. They are essential tools in simplifying expressions and solving trigonometric equations. In the given exercise, understanding the identity for secant is key.
The secant function, denoted as \( \sec x \), is defined as the reciprocal of the cosine function: \( \sec x = \frac{1}{\cos x} \). This identity is crucial because it allows us to rewrite the expression \( \sin x \sec x \) in a simpler form. By multiplying \( \sin x \) with \( \frac{1}{\cos x} \), we simplify the expression to \( \tan x \) by another fundamental identity: \( \tan x = \frac{\sin x}{\cos x} \). Thus, understanding these identities allows us to see that different expressions can essentially convey the same meaning.

Equivalent Expressions

Equivalent expressions are different expressions that simplify to the same result for all values of any variables involved. Recognizing equivalent expressions is crucial in mathematics because it allows you to simplify complex problems or verify the correctness of equations. In our exercise, we are examining whether the expressions \( y_{1} = \sin x \sec x \) and \( y_{2} = \tan x \) are equivalent.
By using the trigonometric identity for secant, we can rewrite \( \sin x \sec x \) as \( \sin x \times \frac{1}{\cos x} \). Simplifying this gives us \( \tan x \), which confirms that both \( y_{1} \) and \( y_{2} \) simplify to the same expression.
This understanding ensures that both formulas are indeed equivalent, clearly demonstrating how mathematical expressions can take different forms yet still represent the same relationship.

Graphical Analysis

Graphical analysis involves comparing graphical representations of functions to understand their behavior and verify equivalence of expressions. In this exercise, we graph the equations \( y_{1} = \sin x \sec x \) and \( y_{2} = \tan x \) using a graphing utility. This provides a visual means to verify their equivalency.
When plotting these functions, you should look for the graphs to overlap completely across their domains. If they do, it implies that the two expressions describe the same function. Visual inspection can often reveal equivalencies that may not be immediately obvious through algebraic manipulation alone.

• If the graphs coincide perfectly, it confirms the equivalency visually.
• Ensure to use the same scale to avoid misinterpretation.

Graphical analysis complements algebraic methods by providing a straightforward visual check.

Algebraic Verification

Algebraic verification is a method of proving equivalency by manipulating expressions using mathematical rules and identities. For our given problem, algebraic verification provides a clear path to confirm that \( y_{1} \) and \( y_{2} \) are equivalent.
We start with the trigonometric identity for secant: \( \sec x = \frac{1}{\cos x} \). By expressing \( \sin x \sec x \) as \( \frac{\sin x}{\cos x} \), it transforms to \( \tan x \), utilizing the identity \( \tan x = \frac{\sin x}{\cos x} \).
This step-by-step transformation confirms that \( y_{1} \) simplifies directly to \( y_{2} \), thus proving algebraically they are equivalent.

• This algebraic step, although symbolic, ensures without reliance on graphical tools alone.
• These transformations leverage identity rules, offering a rigorous proof.

Algebraic verification thus solidifies the conclusions drawn from visual or intuitive approaches.