1. Warm Up

2. Mastery Objectives Convert degree measures of angles to radian measures, and vice versa. Use angle measures to solve real-world problems.

3. Vocabulary Vertex: the common endpoint that 2 noncollinear rays share Initial side: starting position of the ray Terminal side: ray’s position after the rotation Standard position: an angle with its vertex at the origin and its initial side along the positive x-axis.

4. Radian Measure

5. Measure off radians Use included circle and measure off radians

6. The ratio s/r is unitless Draw the following: 1 rotation½ rotation ¼ rotation1/6 rotation

7. Conversion Rules In conversions: Degrees to radians are usually fractions. Write as a fraction and simplify Radians to degrees, replace the π with 180 and simplify. If no units of angle measure are specified, radian measure is implied. If degrees are intended, the degree symbol (º) must be used

9. Write 135° in radians as a multiple of π. Also, write the angle in standard position. Answer:

10. Write –30° in radians as a multiple of π. Also, write the angle in standard position. Answer:

11. Answer:120° Write in degrees. Also, write the angle in standard position.

12. Answer: –135° Write in degrees. Also, write the angle in standard position.

13. Write 150 o in radians as a multiple of π. Also, write the angle in standard position. A. B. C. D.

14. Conterminal Angles Coterminal angles: 2 angles that have the same initial and terminal sides but different measures. To find a positive coterminal angle, add 360º or 2π. To find a negative coterminal angle, subtract 360º or 2π.

15. Identify all angles that are coterminal with 80°. Then find and draw one positive and one negative angle coterminal with 80°. Answer: Sample answers: 440 o, –280 o

16. Identify one positive and one negative angle that are coterminal with. Then find and draw one positive and one negative angle coterminal with. Answer: Sample answer:

17. Identify one positive and one negative angle coterminal with a 126 o angle. A.486°, –234° B.54°, –126° C.234°, –54° D.36°, –486°

18. Arc Length

19. Find the length of the intercepted arc in a circle with a central angle measure of and a radius of 4 inches. Round to the nearest tenth. The length of the intercepted arc is or about 4.2 inches.

20. Find the length of the intercepted arc in a circle with a central angle measure of 125° and a radius of 7 centimeters. Round to the nearest tenth. Answer:15.3 cm

21. A.2.4 centimeters B.4.7 centimeters C.28.3 centimeters D.45° Find the length of the intercepted arc in a circle with radius 6 centimeters and a central angle with measure.

22. Linear and Angular Speed The formula for arc length can be used to analyze circular motion. Linear speed is a rate of circular motion. Linear speed: The rate at which an object moves along a circular path. Measured in units like miles per hour Angular speed: The rate at which the object rotates about a fixed point is called its angular speed. Measured in units like revolutions per minute or radians per minute

23. A typical vinyl record has a diameter of 30 cm. When played on a turn table, the record spins at revolutions per minute. Find the angular speed, in radians per minute, of a record as it plays. Round to the nearest tenth. Answer:209.4 radians per minute

24. Angular speed Answer:209.4 radians per minute Therefore, the angular speed of the record is or about 209.4 radians per minute. Because each rotation measures 2 π radians, revolutions correspond to an angle of rotation

25. A typical vinyl record has a diameter of 30 cm. When played on a turn table, the record spins at revolutions per minute. Find the linear speed at the outer edge of the record as it spins, in centimeters per second. Answer:about 52.4 cm/s

26. Linear Speed s = r  Simplify. A rotation of revolutions corresponds to an angle of rotation

27. Example 5 Answer:about 52.4 cm/s Find Angular and Linear Speeds Use dimensional analysis to convert this speed from centimeters per minute to centimeters per second. Therefore, the linear speed of the record is about 52.4 centimeters per second.

28. Find the angular speed of a carousel in radians per minute if the diameter is 6 feet and it rotates at a rate of 10 revolutions per minute. A.31.4 radians per minute B.62.8 radians per minute C.188.5 radians per minute D.377.0 radians per minute

29. Area of a Sector Sector: region bounded by a central angle and its intercepted arc.

30. Find the area of the sector of the circle. Answer:

31. Therefore, the area of the sector is or about 29.5 square meters. Answer: r = 8 and The measure of the sector’s central angle is, and the radius is 5 meters. Area of sector

32. Find the area of the sector of the circle. Answer:

33. Area of sector Answer: Therefore, the area of the sector is or about 33.5 square feet. r = 8 and Convert the central angle measure to radians. Then use the radius of the sector to find the area.

34. Find the area of the sector of the circle. A.7.9 in2 B.15.7 in2 C.58.9 in2 D.117.8 in2

35. Homework Pg. 238: 10-19, 25, 27, 28, 33, 34-37, 42, 43, 44, 49, 50