Radian Measure, Arc Length and Circular Motion 16 Radian Measure, Arc Length and Circular Motion Click on the computer image at the bottom right for a direct web link to an interesting Wikipedia Math Site. Prepared by: Richard Mitchell Humber College

16.1 - Radian Measure 16.1-Radian Measure.

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. BLANK BLANK

16.1-EXAMPLE 1 CONVERSION FACTOR: One way to convert from degrees to radians is to use the conversion factor 1 radian = 57.2960 . In this example, 47. 60 / 57.2960 = 0.831 radians. Convert 47.60 (Decimal Degrees) to radians and revolutions. 16.1-Radian Measure.

16.1-EXAMPLE 2 CONVERSION FACTOR: One way to convert from radians to degrees is to use the conversion factor 1 radian = 57.2960 . In this example, 1.8473 x 57.2960 = 105.840. Convert 1.8473 radians to degrees (Decimal Degrees) and revolutions. 16.1-Radian Measure.

16.1-EXAMPLE 3 CONVERSION FACTOR: One way to convert from radians to degrees is to use the conversion factor 1 radian = 57.2960 . In this example, 1.837520 x 57.2960 = 105.28210. Convert 1.837 520 radians to degrees0, minutes’ and seconds”. 16.1-Radian Measure.

16.1-EXAMPLE extra CONVERSION FACTOR: One way to convert from degrees to radians is to use the conversion factor 1 radian = 57.2960 . In this example, 28.293610 / 57.2960 = 0.493817 radians. Convert 280 17’ 37” (DMS) to decimal degrees (DD) and radians. 16.1-Radian Measure.

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.

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.

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

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

16.1-EXAMPLE 10 If θ = cot - 1 2.745, find θ in radians to four decimal places. 16.1-Radian Measure. Note: Use RADIAN MODE

16.1-EXAMPLE 10 optional If θ = cot - 1 2.745, find θ in radians to four decimal places. Note: Use RADIAN MODE 16.1-Radian Measure.

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

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

16.1-EXAMPLE 13 ANGLE: The central angle must always be converted into a radian measure when using a radian formula (46.80 / 57.2960 = 0.8168 radians) 16.1-Radian Measure.

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

16.2 - Arc Length 16.2-Arc Length.

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. BLANK BLANK

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.

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

16.2-EXAMPLE 17 16.2-Arc Length.

16.2-EXAMPLE 20 Town Latitude 43.60 Town Latitude 0.761 rad LATITUDE ANGLE: θ must be in radians. (43.60 / 57.2960 = 0.761 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.

16.2-EXAMPLE extra CENTRAL ANGLE: θ must be in radians. (125/365) x (6.28 ) = 2.1518 rad 16.2-Arc Length.

16.3 - Uniform Circular Motion

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

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.

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.

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.

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