15.2 Unit Circle
When we work with degrees, we define a revolution as
\[
\begin{align}
\revolution &: \Number
\\
\revolution &= 360^{\degrees}.
\end{align}
\]
For differentials, another definition of \( \revolution \) works much better.
Definition.
When writing angles in radians we define \( \revolution \) to be equal to the circumference of the unit circle.
\[
\begin{align}
\revolution &: \Number
\\
\revolution &= 6.282 \ldots
\end{align}
\]
We cannot write \( \revolution \) exactly in decimal notation: \( \revolution \) is an irrational number.
On the unit circle an angle written in radians is equal to the distance traveled along the circle.
Cosine and Sine
We define the cosine and sine functions by looking at coordinates on the unit circle.
Definition.
On the unit circle
\[
\begin{align}
x &= \cos \ang
\\
y &= \sin \ang
\end{align}
\]
where \( \ang : \Number \) is any angle.
Take note that the cosine and sine functions
\[
\begin{align}
x &\mathrel{\overset{\cos}{\leftarrow}} \ang
\\
y &\mathrel{\overset{\sin}{\leftarrow}} \ang
\end{align}
\]
use \( x \) and \( y \) as output variables.
Since the cosine and sine functions describe a point on the unit circle, they satisfy the famous equation
\[
\begin{align}
(\cos \ang)^2 + (\sin \ang)^2 &= 1.
\end{align}
\]
Let's summarize some familiar cosine and sine values by splitting the unit circle into eighths, like a large pizza, and twelfths, like a clock face.
These values can all be found using geometry and algebra. I'd like to encourage the student to get out pen and paper, and find geometric proofs for the cosine and sine values:
\[
\begin{align}
&\cos \biggl(\frac{1}{8} \revolution\biggr) = \sqrt[2]{\frac{1}{2}}
&
&\sin \biggl(\frac{1}{8} \revolution\biggr) = \sqrt[2]{\frac{1}{2}}
\\
&\cos \biggl(\frac{1}{12} \revolution\biggr) = \sqrt[2]{\frac{3}{4}}
&
&\sin \biggl(\frac{1}{12} \revolution\biggr) = \sqrt[2]{\frac{1}{4}}.
\end{align}
\]
Of course the square root of one-quarter is one-half, but we'll write \( \sqrt[2]{1/4} \) in solidarity with the other cosine and sine values.