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

justaguide | College Teacher | (Level 2) Distinguished Educator

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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)