Cosecant is inversely related to the sine function. Dictionary.com defines cosecant as the ratio of the hypotenuse to the side opposite a given angle. If you know what the sine function represents, then you should recognize cosecant as its inverse.
When referencing from the SOHCAHTOA method, you should know that sine (the “S”) is indicated by opposite over hypotenuse. Cosecant flips the ratio, making it hypotenuse over opposite.
Sine = Opposite/Hypotenuse, Cosecant = Hypotenuse/Opposite
So let’s assume the length of a side opposite from a reference angle is 2. The length of that triangle’s hypotenuse, or longest side, is 4. When you divide the lengths, the sine ratio is ½ but the cosecant ratio is 2.
To visually demonstrate a cosecant ratio, draw an x-y coordinate plane on a blank sheet of paper. Just draw the y-axis and positive x-axis.
On the x-axis label “2?” on the far right, indicating the spot with a vertical slash mark. Label “0” where the axes meet on the left. Eyeball a spot between these points, and draw a slash there. Label it “?.”
Next graph the “sin x” function between 0 and 2?, drawing it as a dotted line. Assuming you know how to graph sin x, cosecant x (csc x) will reference from here.
Without getting into plotting points, reference the sine curve at x=?/2. The y-coordinate should be “1” where the graph first peaks. Draw two rays that sprout in opposite but upward directions.
According to the value of csc x, the rightward curve increases its y-value infinitely but does not reach an x-value of ?. The leftward curve points up forever but does not go past x=0. This is true for csc x between x=0 and x=?.
This idea is similar for points between x=? and x=2?. But since csc x is inversely related to sin x, the U-shaped curve points down with sin x. So in this interval, the leftward curve doesn’t quite reach x=?, and the rightward curve moves infinitely closer to 2? without reaching it.