6.1: Angle Measure

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?

Question 1 will be answered today. Question 2 will be answered in 6.2.

Angle Measure

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.

Angles in Standard Position

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.

Length of a Circular Arc

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.

Area of a Circular Sector

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?

Circular Motion

As a point swirls around a circle, there are two different ways we can quantify how "fast" the point is moving:

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.