Pre-Calculus Section 8.1: Angles, Arcs, & Their Measures (including Linear & Angular Speed) 1. The graph of a function is given as follows:
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1 Pre-Calculus Section 8.1: Angles, Arcs, & Their Measures (including Linear & Angular Speed) 1. The graph of a function is given as follows: 4. Example 1 (A): Convert each degree measure to radians: A: 90 B: 180 C: 270 D: 360 E: 30 F: 45 Determine the average rate of change for the function between the indicated values of the variable. a. b. c. d. e. G: 60 H: 0 5. Example 1 (B): Convert each Radian measure to Degrees: A: B: C: D: E: F: 2. What is the average rate of change of the function a. 10 b. 2 c. 4 d. 8 e. 7 between? 3. Relationship between Degrees and Radians G: 0 H: 6. Example 1 (C): Find the degree measure of the 2 rad. a º b º c º d º 7. Example 1 (D): Find the degree measure of the
2 a. 105º b. 90º c. 84º d. 94º 8. Example 1 (E): Find the degree measure of the a. 313 b. 338 c. 356 d Example 2-3 (B): Find an angle between 0º and 360º; that is coterminal with the 10º Example 2-3 (C): Find an angle between 0º and 360º that is coterminal with the 1,628º. 9. Example 1 (F): Find the degree measure of the 2.6 rad Example 2-3 (D): Find an angle between 0º and 360º that is coterminal with the angle 2,764º. 10. Example 1 (G): Convert each degree measure to radians: A: 90 B: 180 C: 270 D: 360 E: 30 F: 45 G: 60 H: Example 2-3 (E): The measure of the angle 270º in standard position is given. Find two positive angles and two negative angles that are coterminal with the given angle. a. 414º, 774º, 306º, 666º b. 610º, 1,000º, 130º, 405º c. 650º, 960º, 110º, 420º d. 630º, 990º, 90º, 450º 11. Look at Pg as well as Standard Position and Coterminal on Pg NOTE 1 rad = and rad 17. Example 2-3 (F): Find an angle between: that is coterminal with the. 12. Example 2-3 (A): Find an angle between 0 and 360 that is coterminal with the 2.
3 18. Example 2-3 (G): The measure of the angle in standard position is given. Find minimal positive angle and maximal negative angles that are coterminal with the given angle. 21. Example 4 (A): Find the length of the arc s in the figure. a. b. c. Please give the answer to one decimal place. d. 22. Example 4 (B): Find the length of an arc that subtends a central angle of 60 in a circle of radius 25 m. 19. Example 2-3 (H): The measure of the angle in standard position is given. Find minimal positive angle and maximal negative angles that are coterminal with the given angle. a. m b. c. d. 20. Length of A Circular Arc: In a circle of radius r, the length of s of an arc that subtends a central angle of radians is: 23. Example 4 (C): If radius R = 12 and angle a = 120º, the length of the arc s in the figure is 51.27: True or False s = r NOTE: An angle subtended by an arc, line or other curve is one whose two rays pass through the endpoints of the arc 24. Example 4 (D): Find the length of the arc s in the figure if radius R = 14 and angle a = 120 º. 25. Example 4 (E): Find the angle in the figure.
4 29. Area of a Circular Sector with a central angle of radians is: A = Please give the answer to one decimal place. 30. Example 5 (A): The area of a circle is 40 cm 2. The area of a sector of this circle that subtends a central angle of rad is 4 cm. True or False 26. Example 4 (D): A central angle in a circle of radius 5 m is subtended by an arc of length 14 m. Find the measure of in degrees and in radians. 27. Example 4 (F): Find the radius r of the circle in the figure. 31. Example 5 (B): The area of a sector of a circle with a central angle of 6 rad is 192 m 2. The radius of the circle is 8 m. True or False 32. Example 5 (C): Find the area of the sector shown in the figure. 28. Example 4 (G): Find the radius of the circle if an arc of length 3 ft on the circle subtends a central angle of 120. ft 33. Example 5 (D): Find the radius of the circle if the area of the sector is 10.
5 37. A radial saw has a blade with a 7-in. radius. Suppose that the blade spins at 1300 rpm. (a) Find the angular speed of the blade in rad/min. (b) Find the linear speed of the sawteeth in ft/s. r = Round the answer to the nearest tenth. Use = Example 5 (E): The area of a sector of a circle with a central angle of 4 rad is 32 m 2 Find the radius of the circle. 38. A winch of radius 3 ft is used to lift heavy loads. If the winch makes 7 revolutions every 15 s, find the speed at which the load is rising. Please give the answer to three decimal places. ft / s 35. Example 5 (F): The area of a circle is 84. Find the area of a sector of this circle that subtends a central angle of. cm The wheels of a car have radius 11 in. and are rotating at 1500 rpm. Find the speed of the car in mi/h. Please round the answer to the nearest tenth. mi / h 36. Angular Speed:, where t = time and is in radians Linear Speed:, where s = arc length 40. To measure the speed of a current, scientists place a paddle wheel in the stream and observe the rate at which it rotates. If the paddle wheel has radius 3.2 m and rotates at 100 rpm, find the speed of the current in m/s. Use = Since and s = r therefore; Therefore if we know the angular speed, then we can find the linear speed: Please round the answer to the nearest hundredth. m / s
6 41. Clarksburg, West Virginia, and Miami, Florida, lie approximately on the same meridian. Clarksburg has a latitude of 39.5 N and Miami, 25.5 N. Find the distance between these two cities. (The radius of the earth is 3960 mi.) Please round the answer to the nearest mile. mi
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