From Chapter 5, we have two burning questions:

- Is there a meaning to the $t$ value?
- Is there a way to derive the terminal points for $t = \frac{\pi}{6}, \frac{\pi}{3}$ by hand?

Let's first define what an angle is.

If a circle of radius 1 is drawn with the vertex of an angle at its center, then the measure of this angle in **radians** is the length of the arc that subtends the angle.

Picture will be drawn in class.

Since $\pi$ radians is a rotation halfway around the circle, we know that $180^\circ = \pi$ rad. Thus we have the following:

\[1 \text{ rad} = \left(\frac{180}{\pi}\right)^\circ \qquad\qquad 1^\circ = \frac{\pi}{180}\text{ rad}\]

- To convert degrees to radians, multiply by $\frac{\pi}{180}$.
- To convert radians to degrees, multiply by $\frac{180}{\pi}$.

- Express $60^\circ$ in radians.
- Express $\frac{\pi}{6}$ rad in degrees.

You may have noticed the **length of the arc** is basically the $t$ value, except the initial side of the angle did not start on the positive $x$-axis. If the initial side is on the positive $x$-axis, we say the angle is in **standard position**.

(picture in class)

Two angles in standard position are **coterminal** if both the initial side and terminal sides lie on top of each other.

- Find two angles that are coterminal with the angle $\theta = 30^\circ$ in standard position.
- Find two angles that are coterminal with the angle $\theta =\frac{\pi}{3}$ in standard position.

The $t$ value in Chapter 5 is arc length in radians. We also have a method to convert $t$ into the subtended angle. What is the relationship between the arc length and the angle?

In a circle of radius $r$ the length $s$ of an arc that subtends a central angle of $\theta$ radians is \[s = r\theta\]

In Chapter 5 we had $r = 1$, giving $s = \theta$. This means the arc length was exactly the angle measure. Here, we are generalizing to any valid value of $r$.

- Suppose a circle has radius 10 meters. An arc is subtended by an angle of $30^\circ$. Find the arc length.
- Suppose a circle has radius 4 meters. A central angle is subtended by an arc length of 6 meters. Find the angle in radians.

In a circle of radius $r$ the area $A$ of a sector with a central angle of $\theta$ radians is \[A = \frac{1}{2}r^2\theta\]

Insight: Rewrite the formula as $A = \frac{\theta}{2}\cdot r^2$. What does it look like now?

Find the area of a sector of a circle with central angle $60^\circ$ if the radius of the circle is three meters.

For dinner, you tried to order one pizza with radius 9 inches. However, the pizza place ran out of pizzas that are six inches, and offered you four pizzas with radius 4 inches each for the same price. Should you take the deal?

**Linear speed**is the rate at which the total distance traveled is changing (think the $t$ value).**Angular speed**is the rate at which the central angle $\theta$ is changing.

Suppose a point moves along a circle of radius $r$ and the ray from the center of the circle to the point traverses $\theta$ radians in time $t$. Let $s = r\theta$ (the arc length) be the distance the point travels in time $t$. Then the speed of the object can be thought of in two different ways:

Angular speed |
$\omega = \frac{\theta}{t}$ |

Linear speed |
$\nu = \frac{s}{t}$ |

A boy spins a stone in a three feet long sling at a rate of 15 revolutions every 10 seconds. Find the angular and linear velocities of the stone in seconds.