Algebra and Trig. I. A point is a location or position that has no size or dimension.


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1 Algeba and Tig. I 4.1 Angles and Radian Measues A Point A A B Line AB AB A point is a location o position that has no size o dimension. A line extends indefinitely in both diections and contains an infinite amount of points. A plane is a flat, smooth suface that extends indefinitely in all diections, and contains an infinite numbe of points and lines. Plane A A B Line Segment AB AB Ray AB AB A B A line segment o segment is pat of a line and stats and stops at distinct points called endpoints. A ay consists of a point on a line and all points of the line on one side of the point. An angle is fomed by two ays (o lines) that have a common endpoint. One ay is called the initial side and the othe the teminal side. An acute angle: (0 <θ<90 ) An obtuse angle: (90 <θ<180 ) A ight angle: (θ=90, otation) A staight angle: (θ=180, otation) 1 P a g e
2 The clock to the left shows the hou hand (initial side) at twelve and the minute hand (teminal side) at 3. These two hands ae ays and they fom an angle. The common endpoint of the two ays is called the vetex of the angle. C Teminal Side θ B A Vetex Initial Side Angles ae often labeled with lowecase lette Geek lettes, such as alpha(α), beta(β), gamma(γ), and theta(θ). An angle is in standad position if  its vetex is at the oigin of a ectangula coodinate system  its initial side lies along the positive xaxis. α θ α is in standad position α is positive θ is in standad position θ is negative 2 P a g e
3 An angle is positive if geneated by counteclockwise otation. An angle is negative if geneated by clockwise otation. Measuing Angles A degee is a unit of measuing angles. It epesents of a complete otation about the vetex. (Thee ae 360 degees (360 ) in a complete otation o cicle). 90 A ight angle is an angle that measues 90 degees (90 ). Two lines that intesect to fom a ight angle ae pependicula lines. A small squae in the cone of the angle is used to indicate a ight angle. 180 A staight line has an angle measue of 180 degees (180 ). A vetical line uns up and down, and a hoizontal line uns left and ight. Angle Names: Acute Angle measues less than 90, but moe than 0 Staight Angle measues 180 Obtuse Angle measues moe than 90, but less than 180 Right Angle measues 90 Factional pats of angles ae measued in minutes and seconds.  One minute, witten 1 is degee: 1 =  One second, witten 1 is degee: 1 = Example 3 P a g e
4 Many calculatos ae able to change fom degeeminutesecond notation (D M S ) to a decimal and vice a vesa. Calculato: 2 nd Apps Measuing Angles in Radians Anothe way to measue an angle is in adians. Teminal Side One adian Initial Side One adian is the measue of the cental angle of a cicle that intecepts an ac equal to the lengths to the adius of the cicle. (the adius of the cicle is ) A cental angle is an angle whose vetex is at the cente of the cicle. β We find the length of an angle in adians by dividing the length of the intecepted ac by the adius. γ Radian Measue Conside an ac of length s on a cicle of adius. The measue of the cental angle, θ, that intecepts the ac is 4 P a g e
5 Example Find the adian measue of θ if the ac length is 15 inches and the adius is 6 inches? Example Find the adian measue of θ if the ac length is 42 inches and the adius is 12 inches? Relationship between Degees and Radians We know a full otation aound a cicle has 360 and we know that the cicumfeence of a cicle with adius,, is 2π. Thus the adian measue of a cental angle is the cicumfeence of the cicle divided by the cicles adius,. We use the fomula fo adian measue to find the adian measue of the 360 angle. Because one complete otation measues 360 and 2π adians, 360 = 2π adians Dividing both sides by 2, we have 180 = π adians 5 P a g e
6 Convesion between degees and adians: Using the basic elationship π adians = 180, 2 π adians = To convet degees to adians, multiply degees by  To convet adians to degees, multiply adians by Angles that ae factions of a complete otation ae usually expessed in adian measue as factional multiples of π, athe than decimal appoximations. Fo example θ = athe than using the decimal appoximation θ 1.57 Example Convet each angle in degees to adians Example Convet each angle in adians to degees P a g e
7 Dawing Angles in Standad Position To become comfotable with adian measue, conside angles in standad position. Each oigin is the vetex and each initial side is along the positive xaxis. Think of the teminal side as the side of the angle as evolving aound the oigin. Example Dawing angles in standad position; Note one way to do this is to convet to degees P a g e
8 Teminal Side Radian Measue of Angle Degee Measue of Angle 8 P a g e
9 The gaph below shows what is called the unit cicle. It contains the degee measuements and adian measuements Recall that the xaxis is initial side so when moving counteclockwise the angles ae positive, and when moving clockwise the angles ae negative, so instead of having 330 we would have 30, instead of 315 we would have 45, instead of 300 we would have 60, and so on. Also the adian measues would change also so instead of we would have, instead of we would have, and so on. 9 P a g e
10 Two angles with the same initial sides but possibly diffeent otations ae called coteminal angles. Evey angle has infinitely many coteminal angles. Coteminal Angles Inceasing o deceasing the degee of an angle in standad position by an intege multiply of 360 esults in a coteminal angle. Thus an angle of θ is coteminal with angles of θ ± 360 k, whee k is an intege. Inceasing o deceasing the adian measue of an angle in standad position by an intege multiply of 2π esults in a coteminal angle. Thus an angle of θ adians is coteminal with angles of θ ± 2πk, whee k is an intege. Two coteminal angles fo an angle of θ can be found by adding 360 to θ and by subtacting 360 fom θ. Example Assuming the following angles ae in standad position. Find a positive angle less than 360 that is coteminal with each of the following. 1. a 420 angle 2. a 120 angle 10 P a g e
11 Example Assuming the following angles ae in standad position. Find a positive angle less than 2π that is coteminal with each of the following. 1. a 2. a To find a positive coteminal angle less than 360 o 2π, it is sometimes necessay to add o subtact moe than one multiple of 360 o 2π. 11 P a g e
12 Example Assuming the following angles ae in standad position. Find a positive angle less than 360 o 2π that is coteminal with each of the following. 1. a 2. a 3. a 12 P a g e
13 θ x=ac length The Length of a Cicula Ac Let be the adius of a cicle and θ the nonnegative adian measue of a cental angle of the cicle. The length of the ac intecepted by the cental angle is Example A cicle has a adius a 10 inches. Find the length of the ac intecepted by a cental angle of 120. Example A cicle has a adius a 6 inches. Find the length of the ac intecepted by a cental angle of 45. Expess ac length in tems of π. Then ound you answe to the neaest hundeds. 13 P a g e
14 Linea and Angula Speed Think of a caousel it contains fou cicula ows of animals. As the caousel evolves, the animals in the oute ow tavel a geate distance pe unit time than those in the inne ows. By contast, all animals, egadless of ow, complete the same numbe of evolutions pe unit time. All animals in the fou ows tavel at the same angula speed. Linea and Angula Speed If a point is in motion on a cicle of adius though an angle of θ adians in time t, then its linea speed is, whee s is the ac length given by, and its angula speed is. Example If the had dive in a compute otates at 3600 otations pe minute. Expess the angula speed of a had dive in adians pe minute. (Note: 1 evolution = 2π adians) We can establish a elationship between linea speed and angula speed, by dividing both sides of the ac length fomula, by t Thus, linea speed is the poduct of the adius and the angula speed. 14 P a g e
15 Note we can wite linea speed in tems of angula speed. Recall and and. Example A windmill is used to geneate electicity has blades that ae 10 feet in length. The popelle is otating aound at 4 evolutions pe second. Find the linea speed, in feet pe second of the tips of the blades. 15 P a g e
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