Problem 71 - Precalculus Essentials

Question

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?

Step-by-Step Solution

Answer

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 1: 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.

Step 2: 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.

Step 3: 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.

Step 4: 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.