Angular Measure

1. Angular Measure

2. Angles in Standard Position - Definitions An angle is formed by rotating an initial arm about a fixed point. An angle is said to be in standard position when the initial arm is on the positive x-axis and the vertex is at (0, 0). Positive angles have a counterclockwise rotation. Negative angles have a clockwise rotation. A principal angle is the angle measured from the positive x-axis to the terminal arm. The principal angle is always a positive measure. A reference angle is the angle measured from the closest x-axis to the terminal arm. The reference angle is always an acute angle and is always positive. Coterminal angles are angles that share the same terminal arm. Angle measure + 360 n

3. Sketching Angles in Standard Position Sketch the following angles in standard position. State the principal angle, the reference angle, and one positive and one negative coterminal angle. a) 1500 b) -2600 c) 5600 Principal Angle Reference Angle Coterminal Angles 1000 Principal Angle Reference Angle Coterminal Angles 2000 1500 Principal Angle Reference Angle Coterminal Angles 200 300 800 5600 -1600 5100 -2100 4600 -6200

4. Sketching Angles in Standard Position d) -2200 1400 Principal Angle Reference Angle Coterminal Angles 400 To find all coterminal angles: 5000 -5800 q= 140 + 360n where n is an element of the integers.

5. Angles and Their Measure • What is an angle and what are its various components? • What is standard position and when are two angles coterminal? • How do we measure angles? • How do we compute arc length? • How do we compute linear speed and angular speed?

6. Topics • Angles, initial side, terminal side • Degrees, minutes, seconds, and radians • Coterminal, complementary, and supplementary angles • Arc length • Linear speed and angular speed

7. Angles An angle consists of two rays with a common vertex, together with a direction (either clockwise or counter-clockwise). The side where the angle begins is called the initial side. The side where the angle ends is called the terminal side.

8. Angles

9. Standard Position An angle is in standard position if its vertex is at the origin and its initial side lies on the positive x-axis.

10. Standard Position This figure shows a general angle in standard position.

11. Terminology Angles are often denoted by lower case Greek letters, particularly  (alpha),  (beta),  (gamma),  (theta), and  (phi). A positive angle is an angle which is oriented counter-clockwise. A negative angle is an angle which is oriented clockwise.

12. Terminology Two angles are coterminal if they share the same terminal side when drawn in standard position. We say an angle lies in quadrant I if the terminal side lies in quadrant I. Similarly, we can define angles lying in the other quadrants. We say an angle is quadrantal if its terminal side lies on one of the axes.

13. Measuring Angles • There are two types of units used to measure angles: • Degrees, minutes, and seconds • Radians - We will talk about these later

14. Degrees, Minutes and Seconds One full circle is divided into 360 degrees. Degree measure is denoted by a small superscripted circle, e.g. 74o.

15. Degrees, Minutes and Seconds One degree is divided into 60 minutes. Minutes are noted by a prime, e.g. 14. One minute is divided into 60 seconds. Seconds are noted by a double-prime, e.g. 15.

16. Terminology An acute angle lies in the first quadrant. An obtuse angle lies in the second quadrant. A right angle has measure 90 degrees. A straight angle has measure 180 degrees.

17. Remark Two coterminal angles have measures which differ by any multiple of 3600. To find coterminal angles, simply add or subtract any multiple of 3600

18. More Terminology If an angle has measure , then the complement of this angle has measure 90o-. These two angles are called complementary. If an angle has measure , then the supplement of this angle has measure 180o-. These two angles are called supplementary.