y = csc x is periodic function. The period of y = csc x is 2π. Therefore, we will draw the graph of y = csc x in the interval [-π, 2π].
For this, we need to take the different values of x at intervals of 10°. Then by using the table of natural sines we will get the corresponding values of csc x. Take the values of sin x correct to two place of decimal. The values of csc x for the different values of x in the interval [-π, 2π] are given in the following table.
We draw two mutually perpendicular straight lines XOX’ and YOY’. XOX’ is called the x-axis which is a horizontal line. YOY’ is called the y-axis which is a vertical line. Point O is called the origin.
Now represent angle (x) along x-axis and y (or csc x) along y-axis.
Along the x-axis: Take 1 small square = 10°.
Along the y-axis: Take 10 small squares = 1 unity.
Now plot the above tabulated values of x and y on the co-ordinate graph paper. Then join the points by free hand. The continuous curve obtained by free hand joining is the required graph of y = csc x.
Properties of y = csc x:
(i) The graph of the function y = csc x is not a continuous graph, but consists of infinite number of separate branches, the points of discontinuities are at x = nπ, where n = 0, ±1, ±2, ±3, ±4, ……………... .
(ii) As x passes through any point of discontinuity from the left to the right, the value of csc x suddenly changes from (-∞) to (+ ∞).
(iii) Each branch of the curve approaches continuously the two lines parallel to y-axis at two points of discontinuity of the graph. Such lines are called asymptotes to the curve.
(iv) No part of the graph lies between the lines y = 1 and y = -1, since |csc x| ≥ 1.
(v) The portion of the graph between 0 to 2π is repeated over and over again on either side, since the function y = csc x is periodic of period 2π.