Graph of y = sec x

y = sec x is periodic function. The period of y = sec x is 2π. Therefore, we will draw the graph of y = sec 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 cosines we will get the corresponding values of cos x. Take the values of cos x correct to two place of decimal. The values of cos 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 sec 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 = sec x.

Properties of y = sec x:

(i) The graph of the function y = cos x is not a continuous graph, but consists of infinite number of separate branches, the points of discontinuities are at x = (2n + 1)\(\frac{π}{2}\), where n = 0, ±1, ±2, ±3, ±4, ……………... .

The straight lines parallel to y-axis at these points of discontinuities are asymptotes to the different branches of the curve.

(ii) Comparing cosecant-graph and secant-graph we see that cosecant-graph coincides with secant-graph if the former is shifted to the left through 90° this is due to the fact that cos (90° + x) = sec x.

(iii) No part of the graph lies between the lines = 1 and y = -1, since |sec x| ≥ 1.

(iv) The portion of the graph between 0 to 2π is repeated over and over again on either side, since the function y = sec x is periodic of period 2π.

11 and 12 Grade Math

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