4.1 Radian and Degree Measure

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1 4.1 Radian and Degree Measure Accelerated Pre-Calculus Mr. Niedert Accelerated Pre-Calculus 4.1 Radian and Degree Measure Mr. Niedert 1 / 27

2 4.1 Radian and Degree Measure 1 Angles Accelerated Pre-Calculus 4.1 Radian and Degree Measure Mr. Niedert 2 / 27

3 4.1 Radian and Degree Measure 1 Angles 2 Radian Measure Accelerated Pre-Calculus 4.1 Radian and Degree Measure Mr. Niedert 2 / 27

4 4.1 Radian and Degree Measure 1 Angles 2 Radian Measure 3 Degree Measure Accelerated Pre-Calculus 4.1 Radian and Degree Measure Mr. Niedert 2 / 27

5 4.1 Radian and Degree Measure 1 Angles 2 Radian Measure 3 Degree Measure 4 Applications Accelerated Pre-Calculus 4.1 Radian and Degree Measure Mr. Niedert 2 / 27

6 Angles An angle is determine by rotating a ray about it endpoint. Accelerated Pre-Calculus 4.1 Radian and Degree Measure Mr. Niedert 3 / 27

7 Angles An angle is determine by rotating a ray about it endpoint. The starting position of the ray is the initial side of the angle. Accelerated Pre-Calculus 4.1 Radian and Degree Measure Mr. Niedert 3 / 27

8 Angles An angle is determine by rotating a ray about it endpoint. The starting position of the ray is the initial side of the angle. The position after the rotation is the terminal side of the angle. Accelerated Pre-Calculus 4.1 Radian and Degree Measure Mr. Niedert 3 / 27

9 Angles An angle is determine by rotating a ray about it endpoint. The starting position of the ray is the initial side of the angle. The position after the rotation is the terminal side of the angle. The vertex of the angle is the endpoint of the ray. Accelerated Pre-Calculus 4.1 Radian and Degree Measure Mr. Niedert 3 / 27

10 Angles in Standard Position An angle is said to be in standard position if it situated on a coordinate plane with the vertex located at the origin and the initial side coinciding with the positive x-axis. Accelerated Pre-Calculus 4.1 Radian and Degree Measure Mr. Niedert 4 / 27

11 Angles in Standard Position An angle is said to be in standard position if it situated on a coordinate plane with the vertex located at the origin and the initial side coinciding with the positive x-axis. Accelerated Pre-Calculus 4.1 Radian and Degree Measure Mr. Niedert 4 / 27

12 Positive and Negative Angles Positive angles are generated by counterclockwise rotation. Accelerated Pre-Calculus 4.1 Radian and Degree Measure Mr. Niedert 5 / 27

13 Positive and Negative Angles Positive angles are generated by counterclockwise rotation. Negative angles are generated by clockwise rotation. Accelerated Pre-Calculus 4.1 Radian and Degree Measure Mr. Niedert 5 / 27

14 Coterminal Angles Angles are sometimes labeled with Greek letters such as α (alpha), β (beta), and θ (theta). Accelerated Pre-Calculus 4.1 Radian and Degree Measure Mr. Niedert 6 / 27

15 Coterminal Angles Angles are sometimes labeled with Greek letters such as α (alpha), β (beta), and θ (theta). In the figure below, notice that angles α and β have the same initial side and terminal side. This means that these angles are coterminal. Accelerated Pre-Calculus 4.1 Radian and Degree Measure Mr. Niedert 6 / 27

16 Coterminal Angles Angles are sometimes labeled with Greek letters such as α (alpha), β (beta), and θ (theta). In the figure below, notice that angles α and β have the same initial side and terminal side. This means that these angles are coterminal. Specifically, coterminal angles have different measure, but result in the same angle when in standard position. Accelerated Pre-Calculus 4.1 Radian and Degree Measure Mr. Niedert 6 / 27

17 Radian Measure Definition of Radian One radian is the measure of a central angle θ that intercepts an arc s equal in length to the radius r of the circle. Algebraically, this means that where θ is measured in radians. θ = s r Accelerated Pre-Calculus 4.1 Radian and Degree Measure Mr. Niedert 7 / 27

18 Derivation of Radians One revolution around a circle of radius r corresponds to an angle of 2π radians because θ = s r = 2πr r = 2π radians. Accelerated Pre-Calculus 4.1 Radian and Degree Measure Mr. Niedert 8 / 27

19 Derivation of Radians One revolution around a circle of radius r corresponds to an angle of 2π radians because θ = s r = 2πr r = 2π radians. This means that for each of the angles in the circle can be expressed as a portion (or fraction) of 2π. Accelerated Pre-Calculus 4.1 Radian and Degree Measure Mr. Niedert 8 / 27

20 Quadrants Accelerated Pre-Calculus 4.1 Radian and Degree Measure Mr. Niedert 9 / 27

21 Sketching and Finding Coterminal Angles Example a Sketch the angle θ = 3π and find a coterminal angle. 4 b Sketch the angle θ = 13π and find a coterminal angle. 6 c Sketch the angle θ = 2π 3 and find a coterminal angle. Accelerated Pre-Calculus 4.1 Radian and Degree Measure Mr. Niedert 10 / 27

22 Sketching and Finding Coterminal Angles Practice a Sketch the angle θ = 9π 4 b Sketch the angle θ = 5π 6 and find a coterminal angle. and find a coterminal angle. c Sketch the angle θ = 3π 4 and find a coterminal angle. Accelerated Pre-Calculus 4.1 Radian and Degree Measure Mr. Niedert 11 / 27

23 Complementary and Supplementary Angles Practice If possible, find the complement and supplement of each of the angles below. a π 6 b 5π 6 Accelerated Pre-Calculus 4.1 Radian and Degree Measure Mr. Niedert 12 / 27

24 Radian and Degree Measure (Part 1 of 3) Assignment pg #8-24 even Accelerated Pre-Calculus 4.1 Radian and Degree Measure Mr. Niedert 13 / 27

25 Degrees Degrees are the most frequent way we measure angles and indicated by the symbol. Accelerated Pre-Calculus 4.1 Radian and Degree Measure Mr. Niedert 14 / 27

26 Degrees Degrees are the most frequent way we measure angles and indicated by the symbol. One degree (1 1 ) is equivalent to a rotation of 360 of a complete revolution about the vertex. Accelerated Pre-Calculus 4.1 Radian and Degree Measure Mr. Niedert 14 / 27

27 Converting Between Degrees and Radians Because 2π radians corresponds to one complete revolution, degrees, and radians are related by the equation 360 = 2π rad. Accelerated Pre-Calculus 4.1 Radian and Degree Measure Mr. Niedert 15 / 27

28 Converting Between Degrees and Radians Because 2π radians corresponds to one complete revolution, degrees, and radians are related by the equation 360 = 2π rad. Notice that 360 = 2π rad = 1 = π 180 rad. Accelerated Pre-Calculus 4.1 Radian and Degree Measure Mr. Niedert 15 / 27

29 Converting Between Degrees and Radians Because 2π radians corresponds to one complete revolution, degrees, and radians are related by the equation 360 = 2π rad. Notice that 360 = 2π rad = 1 = π 180 rad. Conversions Between Degrees and Radians To convert degrees to radians, multiply degrees by π rad 180. To convert radians to degrees, multiply radians by 180 π rad. Accelerated Pre-Calculus 4.1 Radian and Degree Measure Mr. Niedert 15 / 27

30 Converting Between Degrees and Radians Because 2π radians corresponds to one complete revolution, degrees, and radians are related by the equation 360 = 2π rad. Notice that 360 = 2π rad = 1 = π 180 rad. Conversions Between Degrees and Radians To convert degrees to radians, multiply degrees by π rad 180. To convert radians to degrees, multiply radians by 180 π rad. Overall, just keep in mind that since 360 = 2π rad then 180 = π rad. Accelerated Pre-Calculus 4.1 Radian and Degree Measure Mr. Niedert 15 / 27

31 Converting Between Degrees and Radians Because 2π radians corresponds to one complete revolution, degrees, and radians are related by the equation 360 = 2π rad. Notice that 360 = 2π rad = 1 = π 180 rad. Conversions Between Degrees and Radians To convert degrees to radians, multiply degrees by π rad 180. To convert radians to degrees, multiply radians by 180 π rad. Overall, just keep in mind that since 360 = 2π rad then 180 = π rad. From here on out, if a unit of measure for an angle is not indicated, then assume the angle is in radians. Degrees will need to be indicated using the symbol from here on out. Accelerated Pre-Calculus 4.1 Radian and Degree Measure Mr. Niedert 15 / 27

32 Converting from Degrees to Radians Practice Convert each of the following from degrees to radians. Express as a multiple of π. a 135 b 540 c 270 Accelerated Pre-Calculus 4.1 Radian and Degree Measure Mr. Niedert 16 / 27

33 Converting from Radians to Degrees Practice Convert each of the following from radians to degrees. Round to the nearest hundredth if necessary. a π 2 b 9π 2 c 2 Accelerated Pre-Calculus 4.1 Radian and Degree Measure Mr. Niedert 17 / 27

34 Degrees-Minutes-Seconds Sometimes you may see angles in degree form expressed as Accelerated Pre-Calculus 4.1 Radian and Degree Measure Mr. Niedert 18 / 27

35 Degrees-Minutes-Seconds Sometimes you may see angles in degree form expressed as This is read as 108 degrees, 16 minutes, and 20 seconds, just like it is with time. We can express this angle in decimal degree form as seen below = Accelerated Pre-Calculus 4.1 Radian and Degree Measure Mr. Niedert 18 / 27

36 Degrees-Minutes-Seconds Sometimes you may see angles in degree form expressed as This is read as 108 degrees, 16 minutes, and 20 seconds, just like it is with time. We can express this angle in decimal degree form as seen below = We can convert an angle already in decimal degree form, such as , as seen below = a little more than min = 7.2min.2 60 s = 12s = Accelerated Pre-Calculus 4.1 Radian and Degree Measure Mr. Niedert 18 / 27

37 Degrees-Minutes-Seconds Sometimes you may see angles in degree form expressed as This is read as 108 degrees, 16 minutes, and 20 seconds, just like it is with time. We can express this angle in decimal degree form as seen below = We can convert an angle already in decimal degree form, such as , as seen below = a little more than min = 7.2min.2 60 s = 12s = Accelerated Pre-Calculus 4.1 Radian and Degree Measure Mr. Niedert 18 / 27

38 Degrees-Minutes-Seconds Sometimes you may see angles in degree form expressed as This is read as 108 degrees, 16 minutes, and 20 seconds, just like it is with time. We can express this angle in decimal degree form as seen below = We can convert an angle already in decimal degree form, such as , as seen below = a little more than min = 7.2min.2 60 s = 12s = Accelerated Pre-Calculus 4.1 Radian and Degree Measure Mr. Niedert 18 / 27

39 Degrees-Minutes-Seconds Sometimes you may see angles in degree form expressed as This is read as 108 degrees, 16 minutes, and 20 seconds, just like it is with time. We can express this angle in decimal degree form as seen below = We can convert an angle already in decimal degree form, such as , as seen below = a little more than min = 7.2min.2 60 s = 12s = Accelerated Pre-Calculus 4.1 Radian and Degree Measure Mr. Niedert 18 / 27

40 Degrees-Minutes-Seconds Sometimes you may see angles in degree form expressed as This is read as 108 degrees, 16 minutes, and 20 seconds, just like it is with time. We can express this angle in decimal degree form as seen below = We can convert an angle already in decimal degree form, such as , as seen below = a little more than min = 7.2min.2 60 s = 12s = Accelerated Pre-Calculus 4.1 Radian and Degree Measure Mr. Niedert 18 / 27

41 Degrees-Minutes-Seconds Sometimes you may see angles in degree form expressed as This is read as 108 degrees, 16 minutes, and 20 seconds, just like it is with time. We can express this angle in decimal degree form as seen below = We can convert an angle already in decimal degree form, such as , as seen below = a little more than min = 7.2min.2 60 s = 12s = Accelerated Pre-Calculus 4.1 Radian and Degree Measure Mr. Niedert 18 / 27

42 Radian and Degree Measure (Part 2 of 3) Assignment Part 1: pg #8-24 even Part 2: pg. 291 #32-46 even, 47-54, odd Accelerated Pre-Calculus 4.1 Radian and Degree Measure Mr. Niedert 19 / 27

43 Arc Length Arc Length For a circle of radius r, a central angle θ intercepts an arc of length s given by s = rθ where θ is measured in radians. Note that if r = 1, then s = θ, and the radian measure of θ equals the arc length. Accelerated Pre-Calculus 4.1 Radian and Degree Measure Mr. Niedert 20 / 27

44 Finding Arc Length Practice A circle has a radius of 4 inches. Find the lenght of the arc intercepted by a central angle of 240. Accelerated Pre-Calculus 4.1 Radian and Degree Measure Mr. Niedert 21 / 27

45 Linear and Angular Speeds Linear and Angular Speeds Consider a particle moving at a constant speed along a circular arc of radius r. If s is the length of the arc traveled in time t, then the linear speed v of the particle is Linear speed v = arc length time = s t. Moreover, if θ is the angle (in radian measure) corresponding to the arc length s, then the angular speed ω of the particle is Angular speed ω = central angle time = θ t. Accelerated Pre-Calculus 4.1 Radian and Degree Measure Mr. Niedert 22 / 27

46 Linear and Angular Speeds Linear and Angular Speeds Consider a particle moving at a constant speed along a circular arc of radius r. If s is the length of the arc traveled in time t, then the linear speed v of the particle is Linear speed v = arc length time = s t. Moreover, if θ is the angle (in radian measure) corresponding to the arc length s, then the angular speed ω of the particle is Angular speed ω = central angle time = θ t. Linear speed measures how fast a particle moves. Accelerated Pre-Calculus 4.1 Radian and Degree Measure Mr. Niedert 22 / 27

47 Linear and Angular Speeds Linear and Angular Speeds Consider a particle moving at a constant speed along a circular arc of radius r. If s is the length of the arc traveled in time t, then the linear speed v of the particle is Linear speed v = arc length time = s t. Moreover, if θ is the angle (in radian measure) corresponding to the arc length s, then the angular speed ω of the particle is Angular speed ω = central angle time = θ t. Linear speed measures how fast a particle moves. Angular speed measures how fast the angle changes. Accelerated Pre-Calculus 4.1 Radian and Degree Measure Mr. Niedert 22 / 27

48 Finding Linear Speed Practice The second hand of a clock is 10.2 centimeters long. Find the linear speed of the tip of this second hand as it passes around the clock face. Accelerated Pre-Calculus 4.1 Radian and Degree Measure Mr. Niedert 23 / 27

49 Finding Angular and Linear Speeds Practice A Ferris wheel with a 50-foot radius makes 1.5 revolutions per minute. a Find the angular speed of the Ferris wheel in radians per minute. b Find the linear speed of the Ferris wheel. Accelerated Pre-Calculus 4.1 Radian and Degree Measure Mr. Niedert 24 / 27

50 Sectors A sector of a circle is the region bounded by two radii of the circle and their intercepted arc. Accelerated Pre-Calculus 4.1 Radian and Degree Measure Mr. Niedert 25 / 27

51 Sectors A sector of a circle is the region bounded by two radii of the circle and their intercepted arc. Area of a Sector of a Circle For a circle of radius r, the area A of a sector of the circle with central angle θ is given by where θ is measured in radians. A = 1 2 r 2 θ Accelerated Pre-Calculus 4.1 Radian and Degree Measure Mr. Niedert 25 / 27

52 Area of a Sector of a Circle Practice A sprinkler on a golf course fairway is set to spray water over a distance of 70 feet and rotates through an angle of 120. Find the area of the fairway watered by the sprinkler. Accelerated Pre-Calculus 4.1 Radian and Degree Measure Mr. Niedert 26 / 27

53 Radian and Degree Measure (Part 3 of 3)Assignment Part 1: pg #8-24 even Part 2: pg. 291 #32-46 even, 47-54, odd Part 3: pg #80-94 even, , Radian and Degree Measure Assignment pg #8-24 even, even, 47-54, odd, even, , Accelerated Pre-Calculus 4.1 Radian and Degree Measure Mr. Niedert 27 / 27

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