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 xaxis which is a horizontal line. YOY’ is called the yaxis which is a vertical line. Point O is called the origin.
Now represent angle (x) along xaxis and y (or cos x) along yaxis.
Along the xaxis: Take 1 small square = 10°.
Along the yaxis: Take 10 small squares = 1 unity.
Now plot the above tabulated values of x and y on the coordinate 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 xaxis.
Step III: Divide the xcoordinate of the points where y = cos x crosses xaxis by a and mark maximum and minimum values of y = c cos ax as c and –c on yaxis.
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 xaxis 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 xcoordinates of the points where it crosses xaxis
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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