Section 9.5 The Unit Circle
The trigonometric functions sine, cosine, and tangent each play a specific role in the geometry of the unit circle. The key concepts in this section are summarized in the video below.
Spend a moment reviewing
Definition 9.1. Each trigonometric function is a certain ratio of
\(x\text{,}\) \(y\text{,}\) and
\(r\text{,}\) where
\((x,y)\) is a point on the circle of radius
\(r \gt 0\) centered at the origin
\((0,0)\text{.}\) In particular,
\(\sin(\theta) = y/r\) and
\(\cos(\theta) = x/r\text{.}\) Rewritting these as,
\begin{gather*}
y = r\sin(\theta)\\
x = r\cos(\theta)
\end{gather*}
we see that the sine and cosine determine the coordinates of points on the circle.
Moreover,
\(\tan(\theta) = y/x\) is the slope of the line formed by the terminal side.
Example 9.28. Find a Point on a Circle.
Using a calculator, estimate the coordinates of the point \((x,y)\) on the circle of radius \(4\) determined by the angle \(207^\circ\) in standard position. Then find the slope of the dashed line formed by the terminal side of the angle.
Solution.
\begin{equation*}
x = r \cos(207^\circ) \approx 4 times (-0.891) = -3.56
\end{equation*}
\begin{equation*}
x = r \sin(207^\circ) \approx 4 times (-0.454) = -1.82
\end{equation*}
\begin{equation*}
m = \tan(207^\circ) \approx 0.510
\end{equation*}
If we restrict to the unit circle with \(r=1\text{,}\) we obtain a very direct interpretation of the three trigonometric functions.
Theorem 9.29. Unit Circle Trigonometry.
Let
\(\theta\) be an angle in standard position. Let
\(P(x,y)\) be the corresponding point on the unit circle
\(x^2+y^2 =1\) determined by the terminal ray of
\(\theta\text{.}\) Then
\begin{equation*}
\cos(\theta) = \text{$x$-coordinate of $P$}
\end{equation*}
\begin{equation*}
\sin(\theta) = \text{$y$-coordinate of $P$}
\end{equation*}
\begin{equation*}
\tan(\theta) = \text{slope of terminal ray}
\end{equation*}
Example 9.30.
Sketch an angle of 2 (radians) and use a calculator to determine the coordinates of the point it determines on the unit circle.
