# On the same axes, sketch the graph of y = sin x and y= cos x . Find all the points of intersection.

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### 1 Answer

It isn't possible for me to sketch the graphs that you require here. The points of intersection can be determined in the following way.

To find the points of intersection of the x-axis with y = sin x, equate y = sin x = 0. We get x = arc sin 0 = 0 and pi. As the sine function is periodic. The graph intersects the x-axis at all points given by (2*n*pi , 0) and (pi + 2*n*pi , 0)

Similarly the graph y = cos x intersects the x-axis at (pi/2 + 2*n*pi, 0) and (3*pi/2 + 2*n*pi , 0)

y = sin x intersects the y-axis at (0 , 0) and y = cos x intersects the y-axis at (0, 1).

The two graphs intersect each other at x corresponding to sin x = cos x

=> tan x = 1

=> x = arc tan 1

=> x = pi/4

The points of intersection of the two graphs are (pi/4 + 2*n*pi, 1/sqrt 2) and (5pi/4 + 2*n*pi , -1/sqrt 2)

**y = sin x intersects the x-axis at (2*n*pi , 0) and (pi + 2*n*pi , 0). y = cos x intersects the x-axis at (pi/2 + 2*n*pi, 0) and (3*pi/2 + 2*n*pi , 0). **

**y = sin x intersects the y-axis at (0 , 0) and y = cos x intersects the y-axis at (0, 1). **

**The points of intersection of y = sin x and y = cos x are (pi/4 + 2*n*pi, 1/sqrt 2) and (5pi/4 + 2*n*pi , -1/sqrt 2)**