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Chapter 4: Problem 71

Explain how to use a sine curve to obtain a cosecant curve. Why can the same procedure be used to obtain a secant curve from a cosine curve?

Short Answer

Expert verified
The sine curve can be used to obtain a cosecant curve and the cosine curve can be used to obtain a secant curve due to the reciprocal relationship between these pairs of trigonometric functions. The reciprocal function has a value at each x of the original curve that is equal to 1 divided by the y-value of the original curve, and is undefined wherever the original function crossed the x-axis.

Step by step solution

01

Understanding Sine and Cosecant

The sine and cosecant functions are reciprocals of each other. This means that wherever the sine function has a value, the cosecant function will have a value equal to 1 divided by that sine value. Thus, when \( \sin(x) = 1 \), \( \csc(x) = 1 \), and when \( \sin(x) = 0 \), \( \csc(x) \) is undefined, because division by zero is undefined. As you move around the sine curve, the cosecant curve will mirror these relationships.
02

Sine to Cosecant Curve

To draw the cosecant curve: For every x-coordinate on your sine curve, draw the value that is the reciprocal of the y-coordinate on the cosecant curve. You'll notice the cosecant curve has asymptotes or 'gaps' wherever the original sine curve crossed the x-axis, because you can't have division by zero.
03

Understanding Cosine and Secant

The relationship between cosine and secant is similar to that of sine and cosecant. The cosine and secant functions are also reciprocals of each other. So, for example, when \( \cos(x) = 1 \), \( \sec(x) = 1 \), and when \( \cos(x) = 0 \), \( \sec(x) \) is undefined.
04

Cosine to Secant Curve

Similarly, to draw the secant curve: For every x-coordinate on your cosine curve, draw the value that is the reciprocal of the y-coordinate on the secant curve. The secant curve will have asymptotes wherever the original cosine curve crossed the x-axis.

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Most popular questions from this chapter

Use a graphing utility to graph each pair of functions in the same viewing rectangle. Use a viewing rectangle that shows the graphs for at least two periods. $$y=-2.5 \sin \frac{\pi}{3} x \text { and } y=-2.5 \csc \frac{\pi}{3} x$$

Graph one period of each function. $$y=-\left|2 \sin \frac{\pi x}{2}\right|$$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. When graphing one complete cycle of \(y=A \sin (B x-C)\) I find it easiest to begin my graph on the \(x\) -axis.

Graph \(f, g,\) and \(h\) in the same rectangular coordinate system for \(0 \leq x \leq 2 \pi .\) Obtain the graph of h by adding or subtracting the corresponding \(y\) -coordinates on the graphs of \(f\) and \(g\) $$f(x)=\cos x, g(x)=\sin 2 x, h(x)=(f-g)(x)$$

Determine the amplitude, period, and phase shift of each function. Then graph one period of the function. $$y=3 \cos (2 \pi x+4 \pi)$$
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