1. Chapter 4-1: Measures of Angles as Rotations

2. Review… Angle: The union of two rays which are its sides with the same vertex or endpoint. Angle: The rotation of one side either clockwise or counterclockwise a certain # of degrees around a certain fixed point (vertex). ***An angle measure tells you TWO things: It’s Magnitude (size) and it’s Direction.

3. Clockwise vs. Counterclockwise Rotation: This is where the angle ALWAYS begins. (we will discuss this more later…) 70° Counterclockwise rotation is positive magnitude. -70° Clockwise rotation is negative magnitude.

4. Other units of Measure for Angles: 1) Revolution: - A revolution is a fraction of the full circle or 360°. So 1 revolution = 360°. - Half a revolution = 180°, and so on… Example: Find 1/8 of a counterclockwise revolution: Find 1/3 of a clockwise revolution:

5. Investigation: Q: Can a particular rotation have more than one magnitude? Q: So… how can you find another magnitude for a particular rotation? A: Add or subtract 360° 90 ° -270 °

6. Examples: Draw an angle representing 5/8 of a clockwise revolution, then find 3 other magnitudes for this rotation.

7. Other units of measure continued: 2) Radian Measures: Let’s investigate… So, a Radian is a fraction of the Circumference with a radius of 1. Basic Conversions: 2π = 360° π = 180° π/2 = 90° We use the easy unit conversion of π to find other common angle conversions from special right triangles: 45° = 30° = 60° =

8. Converting angle measures to different units of measure: 360° = 1 Revolution = 2π Radians When converting from one unit of measure to another, set up a proportion between the two units of measure… Example: Convert 80° to exact radians **** FIRST: Ask yourself what 2 of the 3 units are you using in this example!!! -Next, write the basic unit conversion between them. -Finally, finish the proportion with the given information.

9. More examples: -Convert –π/9 to revolutions: (Radians can also be approximate: 2π ≈ 6.28) -How many revolutions is equal to 6 radians?

10. IMPORTANT!!! When using a number that represents degrees, you must always use a degree symbol: ° If you see a π in the angle measure, it is always an exact radian measure. If you don’t see any symbol, it is assumed to be approximate radians.

11. Pulling it all together… 1) Draw an angle with a rotation of 11π/6 2) Give two other same unit magnitudes for this rotation (just think about how we did this when we were in degrees!) 3) Convert this angle to both revolutions and degrees.