2.2 The Unit Circle

Introduction

Radians and degrees

Special points on the unit circle

Special points and arc length in non-unit circles

In the following figure there are 24 equally spaced points on the unit circle. Since the circumference of the unit circle is $2\pi$, each of the points is $\frac{1}{24} \cdot 2\pi = \frac{\pi}{12}$ units apart (traveled along the circle). Thus, the first point counterclockwise from $(1,0)$ corresponds to the distance $t = \frac{\pi}{12}$ traveled along the unit circle. The second point is twice as far, and thus $t = 2 \cdot \frac{\pi}{12} = \frac{\pi}{6}$ units along the circle away from $(1,0)$.

Label each of the subsequent points on the unit circle with the exact distance they lie counter-clockwise away from $(1,0)$; write each fraction in lowest terms.

$30^\circ$

$\frac{2\pi}{3}$ radians

$\frac{5\pi}{4}$ radians

$240^\circ$

$17^\circ$

$2$ radians

For the $45^\circ$-$45^\circ$-$90^\circ$ triangle with legs of length $x$ and $y$ and hypotenuse of length $1$, what does the fact that the triangle is isosceles tell us about the relationship between $x$ and $y$? What are their exact values?

Now consider the $30^\circ$-$60^\circ$-$90^\circ$ triangle with hypotenuse of length $1$ and the longer leg (of length $x$) lying along the positive $x$-axis. What special kind of triangle is formed when we reflect this triangle across the $x$-axis? How can we use this perspective to determine the exact values of $x$ and $y$?

Suppose we consider the related $30^\circ$-$60^\circ$-$90^\circ$ triangle with hypotenuse of length $1$ and the shorter leg (of length $x$) lying along the positive $x$-axis. What are the exact values of $x$ and $y$ in this triangle?

We know from the conversion factor from degrees to radians that an angle of $30^\circ$ corresponds to an angle measuring $\frac{\pi}{6}$ radians, an angle of $45^\circ$ corresponds to $\frac{\pi}{4}$ radians, and $60^\circ$ corresponds to $\frac{\pi}{3}$ radians.

Use your work in (a), (b), and (c) to label the noted point in each of the three respective figures with its exact coordinates.

In a circle of radius $11$, the arc length intercepted by a central angle of $\frac{5\pi}{3}$.

In a circle of radius $3$, the central angle measure that intercepts an arc of length $\frac{\pi}{4}$.

The radius of the circle in which an angle of $\frac{7\pi}{6}$ intercepts an arc of length $\frac{\pi}{2}$.

The exact coordinates of the point on the circle of radius $5$ that lies $\frac{25\pi}{6}$ units counterclockwise along the circle from $(5,0)$.