# Cosine graphs

Cosine graphs has a period of 2π. So it is sufficient to draw the graph in the interval [0,2π]. We first draw the graph in the interval of [,π/2]. For this we use the table given below and cosine (cos x) is a decreasing function in this interval.
 x 0 30 45 60 90 cos x 1 √3/2 1/√2 1/2 0

Now, to draw the graph of y = cos x in the interval pf [Π/2,Π] we use the relation
cos(Π-x)= - cosx
cos(Π + x) = -cos x
If we take the values of x from 0 to Π/2 then the values of cos x will be positive and the graph will be above X-axis.
If we take the values of x from Π/2 to Π then the values of cos x between Π/2 and Π are negative so the graph between Π/2 to Π is below X-axis.
The cosine graphs looks like
this.In order to draw the graph of y=cos(x) we will use the following steps :
1) Draw a Y-axis with 0,1,-1 ...on it.
2) From the origin draw an X-axis.
a) if you want a graph in π then mark the points π/2, π,3π/2, 2π etc.
As we know that cos(π/2) = 0 so the the cosine curve will intersect at π/2,3π/2, etc.
b) If you want the graph on the number line then mark the points 1,2,3,4...and so on.
The cosine curve will cut the X-axis at 1.57,4.71,etc.
In short in the graph, the value of 1.57 on the X-axis represents $90^{0}$ and 4.71 is equivalent to $270^{0}$ or 2π.
3) y = a cos(x) the amplitude 'a' is 1 so the curve will be up to (0,1). If y = 2 cos(x) then the amplitude will be 2, so the curve will be up to (0,2).

## Cosine graphs

Draw the graph of y = 2 cos(2x)
solution :
Step 1 : Draw Y-axis and mark the the points 0,2 and -2 as the amplitude for y = cos(x) graph is 2. Draw the x-axis from 0 and mark the points π/2,π,3π/2...etc.
Step 2: As cos(0) = 1 so cosine graph starts from the amplitude.
And cos(0) = 1 which is maximum for this particular graph since the amplitude is 2 so the cosine curve is up to [0,2].
similarly cos(π) = -1 which is maximum along the negative Y-axis so the cosine curve is up to [π,-2].
Note : For y = 2 cos x, the amplitude is 2 and cos(0) = 2 so the cosine curve is up to [0,2] on positive Y-axis and [π,-2] on negative Y-axis.
Since the cos x is decreasing function, we obtain the graph of y=cos x in the interval of [0,π]. We draw the graph of y = cos x by using the fact that cos(π- x) = -cos x . So finally, we draw the graph of y= 2 cos x between π and 2π is the mirror image in X-axis of its graph between 0 to 2π.