In order to understand why cos(0) equals 1, we must first understand how we define the cosine of any given angle.
Suppose that we are given an angle θ and we wish to find the value of cos(θ).
Definition of cos(θ):
Consider the unit circle (circle with radius 1) in the XY – coordinate plane with the center at the origin. Draw a line making an angle θ with the x-axis as shown below.
The line meets the unit circle at point P which has coordinates (x,y) as shown in the diagram below. The cosine of the angle θ is now defined as the x-coordinate of the point P on the unit circle below.
This means that in order to find cos(0o) we must first draw a line at an angle of 0o with the x-axis. This line is indicated by the blue line in the diagram below.
Point P where the blue line meets the line circle has coordinates (1,0). This is because the radius of the unit circle is 1.
As mentioned above the cosine of an angle is given by the x-coordinate of the point P. Since the x-coordinate of the point (1,0) is equal to 1, we see that cos(0).
This explains why the cosine of the angle 0 is always equal to 1.