A circle of radius has a circumference of
.
When measuring angles by degrees (), a full revolution is
. For our trigonometric functions, we use radians as
our arguments.
To convert between degrees and radians, one should find the arc length of
the segment of the unit circle demarked by two radii meeting at an
angle of .
From the equality
we find that corresponds to
radians.
A negative arc length should be interpreted as a distance along the unit circle in the clockwise direction.
In terms of triangles, the sine of an angle is the ratio of the
length of the opposite side by the length of the hypotenuse.
The cosine of the angle is the ratio of the length of the adjacent side by
the length of the hypotenuse.
The point on the unit circle radians counterclockwise from
is
.
If is a positive integer, then one write
for
. Likewise, for the cosine
and other trigonometric functions.
The Pythagorean theorem may be expressed as:
As an arclength of corresponds to a full revolution of the
circle,
Let
. Compute
.