The Sine Wave

As described in Sine & Cosine Definitions, t is the length of an arc on the unit circle. If the sine function is graphed with t on the horizontal axis and sin(t) on the vertical axis, this is the resulting curve:

Important points:

• The curve oscillates between 1 and -1.
• The curve repeats every 2pi.

A look at the unit circle shows the reasons. As t proceeds around the unit circle, sin(t) goes up to 1 then down to -1, and then repeats as t keeps going around the circle. It repeats every 2pi because the circumference of the unit circle is 2pi.

Cosine
The graph of cosine is the same as sine, but shifted by -pi/2.

So sin(t) = cos(t - pi/2). For oscillation, either can be used to the same effect. A very interesting relation between sine and cosine is that the slope of the sine curve at any value of t is cos(t).