Math 140
Class 9
Angles and their measures
What are angles?
What are angles?
Angle is a rotation
Counterclockwise rotations are positive
Clockwise rotations are negative
They can be easily added.
Each angle has an initial side and terminal side
Infinitely many angles have the same initial and terminal sides.
Different pairs of initial and terminal sides can correspond to the same angle.
Standard position
when the initial side is the positive \(x\)-axis.
We say that two angles are co-terminal if their standard positions share the same terminal side.
Measuring Angles
Standard way of measuring:
The measure of an angle is the ratio of the arc length of the rotation and the radius of the rotation.
This is sometimes called the radian measure.
What are the units?
Unit circle
When the radius is 1, then the radian measure is simply the unit-less arc length.
For each angle, there is a point on the unit circle that corresponds to it.
Co-terminal angles correspond to the same point.
Other measure systems
- Turns: The full angle is 1 turn.
- Degrees: The full angle is \(360^\circ\).
- Gons: The full angle is \(400^g\).
Cosine and Sine functions
Definition: If \(t\) is a point on the unit number circle, then \(\cos t\) is its \(x\)-coordinate and \(\sin t\) is its \(y\)-coordinate.
