Experience First

Students will use the sine curve to help with the shape of the cosecant curve.  They should fill in the table to see where the vertical asymptotes are as well as key points and then use their knowledge of the relationship between cosecant and sine to develop the overall shape of the curve.  If they try to draw a “V” shape, have them estimate a few more points on the curve for a more exact shape.  Consider pausing after questions 1 through 5 to make sure students have the correct shape before letting them try to draw the secant curve in number 7.

​

Formalize Later

The most important thing to take away from this lesson is that you can use the sine and cosine curves to help draw the cosecant and secant curves, respectively.  Asymptotes occur where the reciprocal function equals 0, reciprocal trig functions intersect at y = 1 and -1, and have an indirect relationship everywhere else.  Have them write asymptote equations for all possible asymptotes by using “n” since it is impossible to list them all.  For the secant curve, students should be drawing two complete “U” shapes, so they need to go past  to get all 3 positive asymptotes.