Section 7.1 Angles and Their Measure


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1 Secion 7.1 Angles and Their Measure Greek Leers Commonly Used in Trigonomery Quadran II Quadran III Quadran I Quadran IV α = alpha β = bea θ = hea δ = dela ω = omega γ = gamma DEGREES The angle formed by roaing he iniial side exacly once in he counerclockwise direcion unil i ends a he saring poin (i.e., one complee revoluion) is said o be 30 degrees (30 ). (You probably remember from your high school geomery class ha if you go around a circle, you have gone 30.) So, one degree is equal o 1 / 30 of a revoluion. 90 is a quarer of a revoluion (because 90 = ¼ of 30) and 180 is half a revoluion (because 180 = ½ of 30). Obviously, 90 and 180 are quadranal angles. Wha are he oher 2 quadranal angle measuremens? and 7.1 Page 1
2 Example: Draw each angle and sae which quadran i is in OR if i is a quadranal angle. a) 0 b) 120 c) 540 d) 225 RADIANS If you draw an angle whose verex is in he cener of a circle, i is called a cenral angle. The poins where he rays of he angle inersec wih he circle creae an arc. If he radius of he circle is equal o he lengh of he arc, hen he measure of he angle is 1 radian. A circle whose radius is one is called a uni circle. We will use he uni circle A LOT during his class! For a circle of radius r, a cenral angle of θ radians creaes (subends) an arc whose lengh s is equal o he radius of he circle imes he cenral angle. s = r θ (arc lengh equals radius imes he angle ha creaed he arc) **The cenral angle mus be in RADIANS in order o use his formula!** Example: Find he missing quaniy. a) r = fee, θ = 2 radians b) r = meers, s = 8 meers 7.1 Page 2
3 CONVERTING BETWEEN DEGREES AND RADIANS Definiion: 1 revoluion = 2π radians And since we already know ha 1 revoluion = 30, ha means ha 2π radians = 30. Dividing boh sides by 2 gives us: π radians = 180. This equaion gives us our wo conversion formulas: If you are given a radian measuremen, muliply i by 180 rad o find is degree measuremen. If you are given a degree measuremen, muliply i by rad 180 o find is radian measuremen. To help you remember which fracion o muliply by, jus remember ha he unis of measuremen of wha you sar wih have o cancel ou and you wan o be lef wih he uni you are rying o conver o. Example 1: Conver beween radians and degrees. a) 225 = radians rad Cancel Degrees 225 Reduce Fracion rad rad b) 11 degrees Cancel Radians 11 rad rad and 's rad 180 rad Reduce Fracion (30) Example 2: Le s figure ou wha he radian measuremen is for our quadranal angles. a) 0 = radians b) 90 = radians c) 180 = radians d) 270 = radians 7.1 Page 3
4 Some oher commonly used radian angle measuremens are,, and radians. Conver hese angles o heir 4 3 degree measuremens. Wha quadran do hese angles fall in? e) radians = degrees f) radians = degrees 4 Quadran Quadran g) radians = degrees 3 Quadran AREA OF A SECTOR OF A CIRCLE If we go back o he idea of placing he verex of an angle a he cener of a circle (creaing a cenral angle, as menioned earlier), he area creaed by he angle is called a secor. (The shaded area in he circle on he righ is a secor.) The formula for he Area of a secor is, where r is he radius of he circle and θ is he radian measuremen of he cenral angle. As wih he arc lengh formula, he cenral angle mus be in RADIANS in order o use his formula. Example 3: A denoes he area of he secor of a circle or radius r formed by he cenral angle θ. Find he missing quaniy. Think abou he uni of Area!! a) r = fee, θ = 2 radians, A = b) θ = 120, r = 3 meers, A = 7.1 Page 4
5 LINEAR SPEED OF AN OBJECT TRAVELING IN CIRCULAR MOTION Suppose ha an objec moves around a circle of radius r a a consan speed. If s is he disance raveled in ime s around his circle, hen he linear speed, v, of he objec is defined as v. (Noe ha s is a DISTANCE, no a speed. Sudens ofen hink s should be speed because he word speed begins wih he leer s. Bu you will find ha in his class and your fuure calculus classes, s always means eiher posiion or disance. v for velociy is he leer mos commonly used for speed.) As his objec ravels around he circle, suppose ha θ (measured in radians) is he cenral angle swep ou over a period of ime. We define he angular speed, ω, of he objec as. Since we previously learned ha arc lengh, s, is equal o radius imes he cenral angle (s = r θ), hen we can replace s in r he linear speed formula wih r θ, o ge v. Bu since is equal o angular speed, ω, he linear speed formula v simplifies o v r. If we solve for, we ge. r In oher words, he linear speed (lengh per uni ime) equals he radius imes he angular speed (where he angular speed is measured in radians per uni ime). Ofen he angular speed is given in revoluions per uni ime, so you have o conver his o radians, remembering ha 1 revoluion = 2π radians. Example 4: An objec is raveling around a circle wih a radius of 2 meers. If he objec is released and ravels 5 meers in 20 seconds, wha is is linear speed? Wha is is angular speed? Example 5: The radius of each wheel of a car is 15 inches. If he wheels are urning a a rae of 12 revoluions per second, how fas is he car moving? Express your answer in inches per second and also in miles per hour. (5280 fee = 1 mile) 7.1 Page 5
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