1. Radian Measure, Arc Length and Circular Motion16
Radian Measure, Arc Length and Circular Motion
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Prepared by: Richard Mitchell Humber College

2. Radian Measure
16.1-Radian Measure.

3. 16.1-DEFINITIONS Measures BLANK BLANK 57.30
CENTRAL ANGLE: The central Angle θ is one whose vertex is at the centre of the circle.
π/2=900
or 1.57rads
ARC LENGTH: An arc having a length equal to the radius of the circle subtends a central angle of one radian. An arc having a length of twice the radius subtends a central angle of two radians and so on. Thus, an arc with a length of 2π times the radius (the entire circumference) subtends a central angle of 2π radians. Therefore, 2π radians is equal to 1 revolution or 3600.
Arc Length
57.30
Central Angle θ
π=1800
or 3.14rads
2π=3600
or 6.28rads
RADIAN: One Radian (~ 57.30) is where an arc is laid off along a circle having a length equal to the radius of the circle.
16.1-Radian Measure.
3π/2=2700
or 4.71rads
π RADIAN: Radian measures can be expressed in either decimal form or in terms of π.
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4. 16.1-EXAMPLE 1
CONVERSION FACTOR: One way to convert from degrees to radians is to use the conversion factor 1 radian = In this example, / = radians.
Convert (Decimal Degrees) to radians and revolutions.
16.1-Radian Measure.

5. 16.1-EXAMPLE 2
CONVERSION FACTOR: One way to convert from radians to degrees is to use the conversion factor 1 radian = In this example, x =
Convert radians to degrees (Decimal Degrees) and revolutions.
16.1-Radian Measure.

6. 16.1-EXAMPLE 3
CONVERSION FACTOR: One way to convert from radians to degrees is to use the conversion factor 1 radian = In this example, x =
Convert radians to degrees0, minutes’ and seconds”.
16.1-Radian Measure.

7. 16.1-EXAMPLE extra
CONVERSION FACTOR: One way to convert from degrees to radians is to use the conversion factor 1 radian = In this example, / = radians.
Convert ’ 37” (DMS) to decimal degrees (DD) and radians.
16.1-Radian Measure.

8. 16.1-EXAMPLE 4
Express 1350 (Decimal Degrees) in radian measure in terms of π.
To convert an angle from degrees to radians in terms of π, multiply the angle by (π rad/1800) and reduce to lowest terms.
16.1-Radian Measure.

9. 16.1-EXAMPLE 5 Convert 7π/9 radians to decimal degrees (DD).
To convert an angle from π radians to degrees, multiply the angle by (1800/π rad) and reduce to lowest terms.
16.1-Radian Measure.

10. 16.1-EXAMPLES 6 and 9
Use your calculator to verify the following trigonometric ratios.
16.1-Radian Measure.
Note: Use RADIAN MODE

11. 16.1-EXAMPLES 7 and 8 Find sec 0.733 radians to three decimal places.
Note: Use RADIAN MODE
16.1-Radian Measure.

12. 16.1-EXAMPLE 10
If θ = cot , find θ in radians to four decimal places.
16.1-Radian Measure.
Note: Use RADIAN MODE

13. 16.1-EXAMPLE 10 optional
If θ = cot , find θ in radians to four decimal places.
Note: Use RADIAN MODE
16.1-Radian Measure.

14. 16.1-EXAMPLE 12 Evaluate the following to four significant digits:
Note: Use RADIAN MODE
16.1-Radian Measure.

15. 16.1-DEFINITIONS Areas of Sectors and Segments θ SEGMENT SEGMENT
16.1-Radian Measure.
SECTOR θ

16. 16.1-EXAMPLE 13
ANGLE: The central angle must always be converted into a radian measure when using a radian formula (46.80 / = radians)
16.1-Radian Measure.

17. 16.1-EXAMPLE 14
16.1-Radian Measure.
Note: Use RADIAN MODE

18. Arc Length
16.2-Arc Length.

19. 16.2-DEFINITIONS s Arc Length and Central Angle θ BLANK BLANK
CENTRAL ANGLE: The central angle θ (in radians) is equal to the ratio of the intercepted arc length s and the radius r of the circle. θ s
Central Angle
Arc Length
16.2-Arc Length.
ARC LENGTH: If the arc length s is equal to the radius r, we have θ equal to 1 radian. For other lengths s, the angle θ is equal to the ratio of s to r.
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20. 16.2-EXAMPLE 15
CENTRAL ANGLE: When you are dividing s by r to obtain θ , s and r must have the same units. The units cancel leaving θ as a dimensionless ratio of two lengths. Thus the radian is not a unit of measure like the degree or inch, although we usually carry it along as if it were.
16.2-Arc Length.

21. 16.2-EXAMPLE 16 CENTRAL ANGLE: θ must be in radians.
16.2-Arc Length.

22. 16.2-EXAMPLE 17
16.2-Arc Length.

23. 16.2-EXAMPLE 20 Town Latitude 43.60 Town Latitude 0.761 rad
LATITUDE ANGLE: θ must be in radians.
(43.60 / = radians)
16.2-Arc Length.
Latitude is the angle (measured at the earth’s centre) between a point on the earth and the equator. Longitude is the angle between the meridian passing through a point on the earth and the principal (or prime) meridian passing through Greenwich, England.

24. 16.2-EXAMPLE extra CENTRAL ANGLE: θ must be in radians.
(125/365) x (6.28 ) = rad
16.2-Arc Length.

25. 16.3 - Uniform Circular Motion

26. 16.3-DEFINITIONS
Angular Velocity, Angular Displacement and Linear Speed
= Angular Velocity
= Angular Displacement
SEGMENT
16.3-Uniform Circular Motion.
SECTOR

27. 16.3-EXAMPLE 21 TIME: t and ω must be consistent.
ω = (1800 rev/min) x (1 min/60 s) = 30 rev/s
16.3-Uniform Circular Motion.

28. 16.3-EXAMPLE 23 ANGLE: θ and ω must be consistent.
θ = (1000 rev) x (2π rad/1 rev) = 6280 radians
16.3-Uniform Circular Motion.

29. 16.3-EXAMPLE 24
ANGULAR VELOCITY: ω can be either in degrees, radians or revolutions per unit of time. In this example, we must convert from radians to revolutions.
(70.8 rad/min) x (1 rev/2π rad) = 11.3 rev/min
16.3-Uniform Circular Motion.

30. Copyright