The unit circle is described by the equation . Any point on the unit circle satisfies this equation.
Let be any real number. Imagine starting at the point on the unit circle and traveling along the circle through an arc length of units (counterclockwise if is positive, clockwise if is negative). Notice that is the radian measure of the angle through which you've turned!
Now let be the point at which you arrived. The sine and cosine of the real number are defined as follows:
Not to get ahead of ourselves, but notice that it must be true that . Do you see why?
The other trigonometric functions are the tangent, secant, cosecant, and cotangent:
The exact values of the trigonometric functions at certain special angles should be memorized! For now, we can read them off our unit circles.
Notice that the sine and cosine functions are defined for all real numbers---there was no restriction on the real number .
On the other hand, the outputs of the sine and cosine functions are the coordinates of points on the unit circle. Therefore,
for all real numbers .
The sine and cosine functions take, as inputs, any real numbers, and they return, as outputs, real numbers between -1 and 1.
Let's go back to the unit circle definitions of the sine and cosine. Imagine that the number is so large () that the entire unit circle is traced more than one time. The points on the unit circle will be retraced, and the values of the sine and cosine function will be repeated. In fact, the values will be repeated every units.
The sine and cosines functions are periodic with period .
for any integer .
The periodicity of the sine and cosine carries over to the other trig functions, but we'll come back to that later.
In addition to periodicity, the symmetry of the unit circle, invokes a kind of symmetry in the trig functions.
The cosine and the secant are even functions:
The sine, cosecant, tangent, and cotangent are odd functions: