back to Trigonometry video lessons

Draw the graph of y = cos x

Look the graph in this video, left side shows the unit circle with radius r = 1. The center of the circle is the origin. OA is a ray whose one end point is the origin and other endpoint is A. OA is the initial side of the angle that overlap with positive x-axis. If ray OA counterclockwise rotate to the B position, the ray OB is the terminal side of the angle AOB. The terminal side of an angle can be in any quadrant. If ray OA counterclockwise rotate, the angle AOB is a positive angle. If ray OA clockwise rotate, the angle AOB is a negative angle. The figure in right side shows a coordinate plane. The number in the x-axis represents the angle AOB rotated from its initial side to its end side. The number in y-axis represents the value of y = cos x.

If OB overlap with OA, then angle AOB = 0. In this case, x = r, y = 0. By the definition, cos AOB = x/r, so cos 0 = x/r = 1.

If OB overlap with positive y-axis, then angle AOB = pi/2. In this case, y = r and x = 0. By the definition, cos AOB = x/r, cos(pi/2) = x/r = 0.

If OB overlay with negative x-axis, then angle AOB = pi. In this case, x = -r and y = 0. By the definition, cos AOB = x/r, cos pi = x/r = -1.

If OB overlap with negative y-axis, then the angle AOB = 3pi/2. In the case, x = 0 and y = -r. By the definition, cos AOB = x/r, cos(3pi/2) = x/r = 0.

If OB overlay with positive x-axis, then the angle AOB = 2pi. In this case, x = r and y = 0. By the definition cos AOB = x/r, cos(2pi) = y/r = 1.

We can select more points in the interval of [0, 2pi] and find the corresponding values of y = cos x, connect those points smoothly to get the graph of y = cos x.

Therefore, the period of y = cos(x) is 2pi.