Problem 71 - Algebra and Trigonometry

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 cosecant curve is obtained from the sine curve by plotting vertical asymptotes where the sine function equals zero and drawing the cosecant curve branches in each non-zero period of the sine function. Similarly, the secant curve is obtained from the cosine curve by plotting vertical asymptotes where the cosine function equals zero and drawing the secant curve branches in each non-zero period of the cosine function. Both procedures are similar because both pairs of functions are reciprocals of each other.

Step 1: Understand Sine Function Graph

Draw a standard graph of the sine function. To get the sine curve, plot the points: \( (0,0), \(\pi/2,1\), \( \pi,0\), \(3\pi/2,-1\), \(2\pi,0\) \), etc. This results in a wave-like curve which oscillates between -1 and 1.

Step 2: Obtain Cosecant Function

The cosecant function is the reciprocal of the sine function. To visualize this, plot vertical asymptotes at the x-values where the sine function is equal to zero. Draw branches of the cosecant curve in each period of the sine function where it's not zero, diverging toward the asymptotes and passing through the maxima and minima of the sine function.

Step 3: Understand Cosine Function Graph

Draw a standard graph of the cosine function. Plot the points: \( (0,1), \(\pi/2,0\), \( \pi,-1\), \(3\pi/2,0\), \(2\pi,1\) \). This produces a curve similar to the sine function graph, but shifted.

Step 4: Obtain Secant Function

The secant function is the reciprocal of the cosine function. Similar to step 2, plot vertical asymptotes at the x-values where the cosine function equals zero. Draw branches of the secant curve in each period of the cosine function that's not zero, diverging toward the asymptotes and passing through the maxima and minima of the cosine function.