Problem 67 - 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. $$ y_{1}=\sin x \csc x, \quad y_{2}=1 $$

Step-by-Step Solution

Answer

Yes, the expressions are equivalent based on both graphical comparison and algebraic verification.

Step 1: Graph the Functions

Graph both functions \(\sin x \csc x\) and \(1\) in the same viewing window of a graphing utility. If the two expressions produce identical graphs, they can be considered equivalent.

Step 2: Verify the Results Graphically

Compare the graphs of the two functions. If the two graphs coincide, the two expressions \(y_{1} = \sin x \csc x\) and \(y_{2} = 1\) are equivalent.

Step 3: Verify the Results Algebraically

Algebraically, observe that \(\csc x\) is the reciprocal of \(\sin x\), i.e. \( \csc x = \frac{1}{\sin x}\). Therefore, the expression \(\sin x \csc x\) can be rewritten as \(\sin x \) * \(\frac{1}{\sin x}\), which equals \(1\). So, this confirms the equivalence of the expressions.

Key Concepts

Graphing Utility

When dealing with trigonometric equations, a graphing utility is a powerful tool. It enables us to visualize functions and compare them within a specific window. This can be especially helpful when checking for equivalency between two expressions. In the provided exercise, using a graphing utility means plotting both expressions, \( y_1 = \sin x \csc x \) and \( y_2 = 1 \), on the same set of axes. Doing so allows one to notice whether the graphs overlap perfectly, indicating that the two expressions are indeed equivalent. Tools like these:

• Facilitate visual learning by providing a concrete image of theoretical concepts.
• Allow for dynamic manipulation of functions to observe changes and similarities.

By analyzing graphs, students can more easily spot equivalencies and key insights that may not be as apparent solely through algebraic work.

Trigonometric Functions

Trigonometric functions, such as sine (\(\sin\)) and cosecant (\(\csc\)), are fundamental in understanding many aspects of mathematics and science. The function \(\sin x\) provides the y-coordinate of a unit circle, which maps out to the varying sine waves we often utilize in problems. It oscillates between -1 and 1. While, \(\csc x\), being the reciprocal of \(\sin x\), is defined as \(\csc x = \frac{1}{\sin x}\). Its behavior is more complex, as it tends toward infinity wherever sine crosses zero.Combining these functions, as seen in the equation \(\sin x \csc x\), yields further insights into trigonometric properties. Here:

• The sine and its reciprocal cosecant simplify beautifully: \(\sin x \csc x = 1\).
• This simplification holds for all values of \(x\) where \(\sin x eq 0\).

Understanding these roles and reciprocal relationships aids in recognizing the inherent equivalency, a crucial skill in trigonometry.

Algebraic Verification

Algebraic verification is crucial when confirming the equivalence of mathematical expressions without solely relying on visual tools like graphs. By revisiting the expressions \(y_1 = \sin x \csc x\) and \(y_2 = 1\) algebraically, we delve into the fundamental trigonometric identities.Here's the breakdown:

• First, note that \(\csc x = \frac{1}{\sin x}\), effectively the reciprocal of the sine function.
• Consequently, multiplying \(\sin x\) by \(\csc x\) simplifies to \(\sin x \times \frac{1}{\sin x} = 1\).

Therefore, algebraically, the expression reduces itself to 1 for all values of \(x\) where \(\sin x\) doesn't equal zero, verifying the equivalence of \(y_1\) and \(y_2\).This approach allows for a rigorous confirmation beyond graphical representation, rooting the correctness of our conclusions in mathematical reason.