Problem 50 - 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. \(y_{1}=\sec ^{2} x-1, \quad y_{2}=\tan ^{2} x\)

Step-by-Step Solution

Answer

The graphs of both expressions overlap, meaning they are equivalent. This is backed by the trigonometric identity that sec^2 x - 1 = tan^2 x.

Step 1: Graphing the functions

Utilize a graphing utility to plot both \(y_{1}=\sec ^{2} x-1, \quad y_{2}=\tan ^{2} x\). Once the functions are plotted, observe if the two functions overlap each other on the graph.

Step 2: Identifying overlapping points

If the graphs of both functions overlap completely, this means they are equivalent as they produce the same outputs for the same x-values.

Step 3: Verifying algebraically

To confirm the results algebraically, use the trigonometric identity \(\sec ^{2} x = 1 + \tan ^{2} x\). If we subtract 1 from both the sides, we get \(\sec ^{2} x - 1 = 1 + \tan ^{2} x - 1\), which simplifies to \(\sec ^{2} x - 1 = \tan ^{2} x\). This matches with the given equations, hence confirming their equivalency.

Key Concepts

Graphing Utilities

Graphing utilities are powerful tools used in mathematics to visually represent equations and their relationships. They help students and professionals quickly see how different functions behave and interact on a coordinate plane. When tackling trigonometric identities such as those in our exercise, graphing utilities can make it easy to check if two expressions produce the same graph.

Here’s why graphing utilities are so essential:

• Visualization: They provide a visual confirmation of the relationships and equivalence between functions.
• Efficiency: Graphing utilities can plot complex functions quickly, saving time on manual graphing.
• Accuracy: They eliminate human error in plotting points, ensuring precise graphs.

In our exercise, graphing the functions \(y_{1} = \sec^{2}x - 1\) and \(y_{2} = \tan^{2}x\) helps to visually confirm their equivalency by checking if their graphs completely overlap.

Trigonometric Functions

Trigonometric functions are fundamental in math, extending into many fields such as physics and engineering. They are derived from the ratios of a right triangle's sides and include functions like sine, cosine, and tangent. In the exercise, we focus on the secant and tangent functions.

Here's a brief rundown:

• Tangent, \(\tan x\): Defined as the ratio of the sine to the cosine functions \( (\tan x = \frac{\sin x}{\cos x})\).
• Secant, \(\sec x\): Defined as the reciprocal of the cosine function \( (\sec x = \frac{1}{\cos x})\).

When we look at the identity \(\sec^{2}x = 1 + \tan^{2}x\), it shows us how these trigonometric functions are intricately related. This identity is extremely useful when verifying the equivalency of trigonometric expressions, as demonstrated in our solution.

Algebraic Verification

Algebraic verification involves the use of identities and algebraic manipulations to confirm the equivalence of mathematical expressions. In our exercise, algebraic verification is a crucial step to prove that \(y_{1} = \sec^{2}x - 1\) is equivalent to \(y_{2} = \tan^{2}x\).

To verify algebraically, we use the identity \(\sec^{2}x = 1 + \tan^{2}x\). By rearranging terms, we show:

• Subtract 1 from both sides of \(\sec^{2}x = 1 + \tan^{2}x\).
• This gives \(\sec^{2}x - 1 = \tan^{2}x\).

This simplification shows that both expressions are equal, confirming their equivalency through algebraic manipulation. Algebraic verification adds a layer of certainty, ensuring that the visual confirmation from the graph aligns with mathematical principles.