Graph of y = cos x

y = cos x is periodic function. The period of y = cos x is 2π. Therefore, we will draw the graph of y = cos 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 cos 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 = cos x.


Steps to draw the graph of y = c cos ax.

Steps I: Obtain the values of a and c.

Step II: Draw the graph of y = cos x and mark the points where y = cos x crosses x-axis.

Step III: Divide the x-coordinate of the points where y = cos x crosses x-axis by a and mark maximum and minimum values of y = c cos ax as c and –c on y-axis.

The graph obtained is the required graph of y = c cos ax.

Properties of y = cos x.

(i) The graph of the function y = cos x is continuous and extends on either side in symmetrical wave form.

(ii) Since the graph of y = cos x intersects the x-axis at the origin and at points where x is an odd multiple of 90°, hence cos x is zero at x = (2n + 1)\(\frac{π}{2}\) where n = 0, ±1, ±2, ±3, ±4, ……………... .

(iii) The ordinate of any point on the graph always lies between 1 and - 1 i.e., - 1 ≤ y ≤ 1 or, -1 ≤ cos x ≤ 1 hence, the maximum value of cos x is 1 and its minimum value is - 1 and these values occur alternately at x = 0, π, 2π,………  i. e., at x = nπ, where n = 0, ±1, ±2, ±3, ±4, ……………...

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


Solved example to sketch the graph of y = cos x:

Sketch the graph of y = 2 cos 3x.

Solution:

To obtain the graph of y = 2 cos 3x we first draw the graph y = cos x in the interval [0, 2n] and then divide the x-coordinates of the points where it crosses x-axis by 3. The maximum and minimum values are 2 and -2 respectively.

Note: Replacing c by 2 and a by 3 in the graph of y = c cos ax, then we get the graph of y = 2 cos 3x






11 and 12 Grade Math

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