Problem 64 - Contemporary Precalculus

Question

In Exercises \(61-64\), use graphs to determine whether the equa. tion could possibly be an identity or is definitely not an identity. $$\frac{\sec t+\csc t}{1+\tan t}=\csc t$$

Step-by-Step Solution

Answer

Answer: Not an identity.

Step 1: Graph of $$y_1 = \frac{\sec t+\csc t}{1+\tan t}$$

To graph the first equation, $$y_1 = \frac{\sec t+\csc t}{1+\tan t}$$, we can use a graphing calculator, or software like Desmos or Geogebra. Alternatively, if you are about to graph the function by rough sketch, remember the following properties: - Sec: secant function is undefined for $$t=90^{\circ} + 180^{\circ}n$$, where n is an integer. - Csc: cosecant function is undefined for $$t = 180^{\circ}n$$, where n is an integer. - Tan: tangent function has asymptotes at $$t=90^{\circ} + 180^{\circ}n$$, where n is an integer.

Step 2: Graph of $$y_2 = \csc t$$

To graph the second equation, $$y_2 = \csc t$$, we can again use a graphing calculator, or software like Desmos or Geogebra. If you graph it manually, remember the following properties: - Csc: cosecant function is undefined for $$t = 180^{\circ}n$$, where n is an integer. - Csc: cosecant function has graph shape like a parabola, with vertical asymptotes at the undefined values.

Step 3: Compare the graphs

Now that we have graphed both equations, compare the graphs of $$y_1$$ and $$y_2$$. If they both match for all values of t, as per their domain, then the given equation is an identity. If the graphs do not coincide for all values of t, then the given equation is not an identity. Based on a visual comparison, you will see that the graphs do not match for all values of t. As a result, we can conclude that the given equation $$\frac{\sec t+\csc t}{1+\tan t}= \csc t$$ is not an identity.

Key Concepts

Graphing Trigonometric Functions

When graphing trigonometric functions, it's important to understand that each function has a unique set of properties and characteristics that affect its graph. To begin with, familiarize yourself with the basic sine and cosine functions, as they are the foundation for other trigonometric functions.

When graphing, pay attention to key features: the period (how often the function repeats itself), amplitude (the height of the peaks), and any asymptotes (lines that the graph approaches but never touches). Additionally, you should be aware of any phase shifts (horizontal moves) and vertical translations (moves up or down).

• Sine and Cosine: Both have a period of \(2\pi\) and an amplitude of 1 (assuming no transformation).
• Tangent: Has a period of \(\pi\) and vertical asymptotes where the function is undefined. The graph is continuous between these asymptotes.
• Secant and Cosecant: These are reciprocals of cosine and sine, respectively. Their graphs have vertical asymptotes corresponding to the zeros of the sine and cosine functions.

By combining these features, you can graph any trigonometric function using either a graphing calculator or by hand using a set of key points and asymptotic behavior.

Secant Function

The secant function, denoted as \( \sec x \), is the reciprocal of the cosine function. This means that \( \sec x = \frac{1}{\cos x} \). To understand the secant function graph, you must first be familiar with the cosine graph.

The key points to graph the secant function are the points where \( \cos x \) is equal to 1 or -1, and those are the maximum and minimum points of the secant function. However, the most distinct feature of the secant graph are its vertical asymptotes; these occur at the values where the cosine function equals zero, since division by zero is undefined.

• The secant function has a period of \(2\pi\).
• Vertical asymptotes exist at \( x = \frac{\pi}{2} + n\pi \), where \( n \) is an integer.
• The graph will mirror the shape of the cosine curve between asymptotes, but will appear in a series of upward and downward facing parabolas.

Understanding these properties will allow you to sketch the secant function accurately.

Cosecant Function

The cosecant function, represented by \( \csc x \), is the reciprocal of the sine function, which can be expressed as \( \csc x = \frac{1}{\sin x} \). Analogous to the secant function and the cosine relationship, knowing the sine graph is essential to graphing the cosecant.

For the cosecant function:

• The zeros of the sine function become the locations of the vertical asymptotes of the cosecant function.
• The period of the cosecant function is the same as the sine function, \(2\pi\).
• Vertical asymptotes are found at \( x = n\pi \), where \( n \) is an integer, corresponding to the zeros of \( \sin x \).
• The graph is characterized by a series of U-shaped and inverted U-shaped curves between these asymptotes.

By paying careful attention to these aspects, the graph of the cosecant function can be drawn accurately for a range of values.

Tangent Function

The tangent function, \( \tan x \), is defined as the ratio of the sine to the cosine functions, or \( \tan x = \frac{\sin x}{\cos x} \). The graph of the tangent function is quite distinct from the sine and cosine graphs.

Some characteristics to consider when graphing the tangent function include:

• The function has a period of \(\pi\), which is shorter than that of sine and cosine functions.
• It has vertical asymptotes wherever the cosine function has zeros, typically at \( x = \frac{\pi}{2} + n\pi \) where \( n \) is an integer.
• The tangent function does not have a maximum or minimum amplitude – its range is all real numbers.
• The graph has a repetitive pattern, with each segment looking similar and fitting between two vertical asymptotes.

Understanding these features and remembering that tangent is undefined at the same places where cosine is zero can aid in successfully graphing the tangent function.