2. RADIAN AND DEGREE MEASURE Objectives: 1. Describe angles 2. Use radian measure 3. Use degree measure 4. Use angles to model and solve real-life problems

3. WHY??? You can use angles to model and solve real-life problems. For example, you can use angles to find the speed of a bicycle

4. ANGLES: Angle = determined by rotating a ray (half-line) about its endpoint. Initial Side = the starting point of the ray Terminal Side = the position after rotation Vertex = the endpoint of the ray Positive Angles = generated by counterclockwise rotation Negative Angles = generated by clockwise rotation

5. An angle is formed by joining the endpoints of two half-lines called rays. The side you measure from is called the initial side. Initial Side The side you measure to is called the terminal side. Terminal Side This is a counterclockwise rotation. This is a clockwise rotation. Angles measured counterclockwise are given a positive sign and angles measured clockwise are given a negative sign. Positive Angle Negative Angle

6. Angle describes the amount and direction of rotation 120°–210° Positive Angle - rotates counter-clockwise (CCW) Negative Angle - rotates clockwise (CW)

7.  = 45° What is the measure of this angle? You could measure in the positive direction  = - 360 + 45° You could measure in the positive direction and go around another rotation which would be another 360°  = 360 + 45°= 405° You could measure in the negative direction There are many ways to express the given angle. Whichever way you express it, it is still a Quadrant I angle since the terminal side is in Quadrant I.  = - 315°

8. RADIAN MEASURE Measure of an Angle = determined by the amount of rotation from the initial side to the terminal side (one way to measure angle is in radians) Central Angles = an angle whose vertex is the center of the circle Radian = the measure of a central angle θ that intercepts an arc “s” equal in length to the radius “r” of the circle Because the circumference of a circle is 2πr units, a central angle of one full rotation (counterclockwise) corresponds to an arc length of s = 2πr. The radian measure of a central angle θ is obtained by dividing the arc length s by r (θ = s/r)

9. initial side terminal side radius of circle is r r r arc length is also r r This angle measures 1 radian Given a circle of radius r with the vertex of an angle as the center of the circle, if the arc length formed by intercepting the circle with the sides of the angle is the same length as the radius r, the angle measures one radian. Another way to measure angles is using what is called radians.

10. QUADRANT IQUADRANT II QUADRANT IIIQUADRANT IV

11. COTERMINAL ANGLES Two angles are coterminal if they have the same initial and terminal sides. The angles 0 and 2π are coterminal. You can find an angle that is coterminal to a given angle θ by adding or subtracting 2π. For positive angles subtract 2π and for negative angles add 2π.

12. Coterminal Angles: Two angles with the same initial and terminal sides Find a positive coterminal angle to 20º Find 2 coterminal angles to Find a negative coterminal angle to 20º

13. COTERMINAL ANGLES EXAMPLE Find two Coterminal Angles (+ and -)

14. Complementary Angles: Two angles whose sum is 90  or (π/2) radians Supplementary Angles: Two angles whose sum is 180  or π radians

15. COMPLEMENTARY & SUPPLEMENTARY ANGLES Two positive angles are complementary if their sum is π/2. Two positive angles are supplementary if their sum is π. Find the Complement and Supplement

16. DEGREE MEASURE Since 2π radians corresponds to one complete revolution, degrees and radians are related by the equations... 360° = 2π rad 180° = π rad

17. CONVERSIONS BETWEEN DEGREES AND RADIANS To apply these two conversion rules, use the basic relationship π = 180° To convert degrees to radians, multiply degrees by To convert radians to degrees, multiply radians by

18. Convert to radians: Convert to degrees:

19. EXAMPLES: Convert from degrees to radians 135° 540° -270° Convert from radians to degrees

20. A Sense of Angle Sizes See if you can guess the size of these angles first in degrees and then in radians. You will be working so much with these angles, you should know them in both degrees and radians.

21. Degrees – Minutes - Seconds With calculators it is easy to denote fractional parts of degrees in decimal form. Historically, however, fractional parts of degrees were expressed in minutes and seconds by using prime (‘) and double prime (“) notations. The graphing calculator can also aid in these conversions. ANGLE Menu To display the ANGLE menu, press 2 nd (ANGLE). The ANGLE menu displays angle indicators and instructions. The Radian/Degree mode setting affects the TI-83 Plus’s interpretation of ANGLE menu entries. Option 4 - DMS (degree/minute/second) This displays answer in DMS format. The mode setting must be Degree for answer to be interpreted as degrees, minutes and seconds. DMS is valid only at the end of a line.

22. 1 degree = 60 minutes 1° = 60 1 minute = 60 seconds 1 = 60  So … 1 degree = _________seconds3600 Express 36  5010  as decimal degrees

23. Express 50.525  in degrees, minutes, seconds 50º +.525(60) 50º + 36.5 50º + 36 +.5(60)  50 degrees, 36 minutes, 30 seconds

24. Arc length s of a circle is found with the following formula: arc lengthradiusmeasure of angle IMPORTANT: ANGLE MEASURE MUST BE IN RADIANS TO USE FORMULA! Find the arc length if we have a circle with a radius of 3 meters and central angle of 0.52 radian. 3  = 0.52 arc length to find is in black s = r  3 0.52 = 1.56 m What if we have the measure of the angle in degrees? First you must convert from degrees to radians before you use the formula. s = r 

26. ARC LENGTH s = rθ (s = arc length, r = radius, θ= central angle measure in radians) EXAMPLES: A circle has a radius of 4 inches. Find the length of the arc intercepted by a central angle of 240°. A circle has a radius of 8 cm. Find the length of the arc intercepted by a central angle of 45°

27. LINEAR AND ANGULAR SPEED Linear speed measure how fast the particle moves, and angular speed measures how fast the angle changes. r = radius, s = length of the arc traveled, t = time, θ = angle (in radians) (distance/time) (turn/time) Ex. 55 mph, 6 ft/sec, 27 cm/min, 4.5 m/sec Ex. 6 rev/min, 360 °/day, 2π rad/hour

28. Example A bicycle wheel with a radius of 12 inches is rotating at a constant rate of 3 revolutions every 4 seconds. a) What is the linear speed of a point on the rim of this wheel?

29. Example A bicycle wheel with a radius of 12 inches is rotating at a constant rate of 3 revolutions every 4 seconds. b) What is the angular speed of a point on the rim of this wheel?

30. Example 2 In 17.5 seconds, a car covers an arc intercepted by a central angle of 120˚ on a circular track with a radius of 300 meters. a) Find the car’s linear speed in m/sec. b) Find the car’s angular speed in radians/sec.

31. Picture it… 300 m

32. Example 2 (a) Linear speed: Note: θ must be expressed in radians

33. (b) Angular speed: ω ≈ 0.12 radians/sec Example 2

34. Example 3 A race car engine can turn at a maximum rate of 12 000 rpm. (revolutions per minute). a)What is the angular velocity in radians per second. Solution a) Convert rpm to radians per second = 1256 radians/s

35. EXAMPLES: The second hand of a clock is 8.4 centimeters long. Find the linear speed of the tip of this second hand as it passes around the clock face. A lawn roller with a 12-inch radius makes 1.6 revolutions per second. Find the angular speed of the roller in radians per second. Find the speed of the tractor that is pulling the roller.