Problem 49 - 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. \(y_{1}=1+\cot ^{2} x, \quad y_{2}=\csc ^{2} x\)

Step-by-Step Solution

Answer

The plots of the two expressions \(y_{1}=1+\cot ^{2} x\) and \(y_{2}=\csc ^{2} x\) overlap everywhere indicating that they are equivalent. This equivalence is further verified by the trigonometric identity \(\cot ^{2} x + 1 = \csc ^{2} x\).

Step 1: Plot the first function \(y_{1}=1+\cot ^{2} x\)

Using a graphing utility, input the expression \(1+\cot ^{2} x\) and generate the graph. Be sure to take note of any relevant points, such as zeros or extrema.

Step 2: Plot the second function \(y_{2}=\csc ^{2} x\)

On the same graphing utility and on the same axes, now input the expression \(\csc ^{2} x\) and generate the second graph. Again, take careful note of any zeros, extrema, or other notable points.

Step 3: Compare the two graphs

By observing the graphs of the two functions, determine if and where they intersect. The points of intersection will also be the values of x for which the two expressions are equivalent.

Step 4: Verify algebraically

Algebraically verify the points of intersection found graphically. Use the properties of trigonometric functions, particularly the identity \(\cot ^{2} x + 1 = \csc ^{2} x\), which directly shows the equivalence of the two original expressions.

Key Concepts

Graphing Utility

A graphing utility is a powerful tool that helps visualize mathematical functions by plotting them on a coordinate plane. These tools, which can be software applications or handheld calculators, are especially useful in understanding the behavior of functions throughout their entire range. By entering the expression of a function, a graphing utility displays its curve, showing features like intercepts, maximum and minimum points, and points of intersection with other functions.

When you have two expressions and you aim to see if they are equivalent, plotting them on a graph using this utility can be revealing. By overlaying both functions, such as in the original exercise with the equations \(y_{1}=1+\cot ^{2} x\) and \(y_{2}=\csc ^{2} x\), one can visually detect if and where the graphs align. If the graphs overlap completely throughout their domain, then the expressions they represent are equivalent across that domain.

Equivalent Expressions

Equivalent expressions are different algebraic expressions that yield the same result for all values of the variables involved. In trigonometry, there are many identities that can make otherwise complex expressions become equivalent through simplification. Identifying these equivalences is crucial in solving trigonometric problems, as they can simplify equations or help verify solutions.

For instance, you can use trigonometric identities to check equivalence. One such identity is \(\cot^2 x + 1 = \csc^2 x\), which directly shows that the expressions \(1 + \cot^2 x\) and \(\csc^2 x\) are indeed equivalent. Recognizing these identities allows for facilitating the solving process and confirms graphically observed results.

Cotangent Function

The cotangent function, denoted as \(\cot(x)\), is the reciprocal of the tangent function. It is calculated as \(\cot(x) = \frac{1}{\tan(x)}\), or, equivalently, \(\cot(x) = \frac{\cos(x)}{\sin(x)}\). Understanding this function is essential because it frequently appears in trigonometric identities and equations.

The graph of the cotangent function differs from the typical sine and cosine curves. It presents vertical asymptotes where the sine function is zero, as \(\cot(x)\) is undefined at these points. Hence, \(\cot^2 x\), which arises in numerous identities, has asymptotical behavior, influencing the graph of expressions like \(1 + \cot^2 x\). Plotting this with a graphing utility shows its unique characteristics and helps explore its intersections with other trigonometric functions.

Cosecant Function

The cosecant function, represented as \(\csc(x)\), is the reciprocal of the sine function. Defined as \(\csc(x) = \frac{1}{\sin(x)}\), it is only defined where \(\sin(x)\) is not zero, leading to vertical asymptotes at those points. This behavior characterizes its graph as quite different from that of standard sine or cosine curves.

By plotting \(\csc^2 x\) on a graphing utility, you can observe the frequency of its oscillations and how it compares to other functions like \(1 + \cot^2 x\). Studying this function graphically aids in comprehending its behavior over different intervals and facilitates comparisons with equivalent expressions. This visual comparison, backed by algebraic verification, ensures a deeper understanding of trigonometric identities and their practical applications.