Trigonometry

Any time we trace a path in the plane, we implicitly define two functions of
time. The first function takes t to the x-coordinate of our path at time t; the
second function takes t to the y-coordinate. Specifying x and y as functions of
t is called parametric graphing. It is more versatile than simply plotting y as a
function of x.
The particular path we are interested in at the moment is the unit circle,
i.e., the circle of radius 1 centered at the origin. The circle is not the graph of
a function, since a single x-coordinate can have more than one corresponding
y-coordinate:
x
y
a single x-value
Nevertheless, the circle can be described, parametrically, by two functions. By
convention, we start at the rightmost point (1, 0) and then move counterclockwise
around the circle at constant speed 1. Then for any given time t, there
is a unique x-coordinate x(t) and a unique y-coordinate y(t), so we have two
functions.
1
It is perhaps natural to ask whether these functions can be described by
formulas. So far in this class, we have used only +, ?, ×, ÷, and exponents and
roots. As it turns out, no finite-length expression using only these operations
can describe either of the two functions x(t) and y(t). Nevertheless, we know
that these two functions do exist. So, we invent new symbols, sin (sine) and cos
(cosine), to describe them.
Definition. Let t be a real number. We define sin t and cos t as follows:
Let (x(t), y(t)) be the point on the unit circle obtained by starting at the
point (1, 0) and traveling a distance of t in the counterclockwise direction.
Then we define
sin t = y(t)
cos t = x(t).
x
y
t
(cos t, sin t)
Note: To make this apply for negative t, we take, e.g., “traveling ?2 units
counterclockwise” to mean “traveling 2 units clockwise.”
This is the same idea as defining the p symbol to solve problems that
otherwise would have no formula for the solution.
2
2 Radians