6.1: Angles and Their Measure

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1 6.1: Angles and Thei Measue Radian Measue Def: An angle that has its vetex at the cente of a cicle and intecepts an ac on the cicle equal in length to the adius of the cicle has a measue of one adian. = 1 adian = adius of the cicle Notice that when you go one adian, on you cove exactly the length of the adius on the cicle. So both ed potions have length. Note: a complete otation in adians is 2π. (The cicumfeence of the unit cicle) Now if we want to go fom adians to degees o fom degees to adians, we will need to use atios to help along with what we know about adians and degees. Fill in the factions below (don t foget to include units!), one complete otation in adians one complete otation in degees = = Ex, Convet to adians o degees whicheve is appopiate ad MTH Wkst

2 Ac Length Pictue illustating the components of ac length is the cental angle is the adius of the cicle The ed potion is the ac (intecepted ac) The ac length is the length of the ed potion Now lets ty to figue out how to calculate the ac length given the adius and cental angle. Fill in the blanks below. Hint: It might be helpful to look at the pictue in Radian Measue If the cental angle is 1 adian and the adius is, then the ac length (ed potion) is. If = 2 adians and the adius is, then the ac length (ed potion) is. If = 3 adians and the adius is, then the ac length (ed potion) is. If = 52 adians and the adius is, then the ac length (ed potion) is. Ac Length fomula The length of an ac, s, intecepted on a cicle of adius by a cental angle of adians is Question: s = Does this fomula wok if is in degees? MTH Wkst

3 Aea of a Secto Now we want to figue out how to find the aea of the shaded egion below. is the cental angle is the adius of the cicle The ed potion is the secto The aea of the secto is the aea of the ed potion We can use popotions to figue out the aea of the secto. We will compae the aea to the cental angle. Fill in the blanks below (use adians), Aea of a secto Aea of a secto = = Aea of a Cicle Complete otation Now solve fo Aea of a secto Aea of a Secto fomula The aea of a secto, A, of a cicle of adius and cental angle of adians is Question: A = Do you get the same fomula if you follow the pocess with in degees? MTH Wkst

4 Rotation and Speed When looking at a otating object, thee ae two diffeent types of speed to conside. Angula Speed Def: The angula speed, ω, measues the speed of otation. is the angle that is tavesed ω = t t is the amount of time that it takes to tavese the angle This fomula falls in line with ou typical definition of speed. If it takes me 2 hous to tavel 120 miles. 120 miles Then I taveled at a speed of oughly = 60 mph. So if I go a distance of adians in say t 2 hous minutes, then I taveled at a ate of ad t min. MTH Wkst

5 Linea (Tangential) Speed Def: The linea (tangential) speed, v, measues the speed of an object in otation at a distance away fom the cente of otation. v = w is the distance fom the object to the cente of otation ω is the angula speed (speed of otation) The linea speed can be thought of the speed of the object (the ed point) if it wee to fly off its path along the cicle and keep going (as illustated below). It might seem a bit weid that thee is a diffeence between angula speed and linea speed. Howeve, we can see the diffeence if we think about the distance to the cente. If two objects (ed point) and (blue point) ae both taveling along a cicula path at the same angula speed (so they cove the same angle in the same amount of time), then one point actually has tavels faste. The blue dot has to tavel faste since it has to cove a geate distance in the same amount of time! Why, because the blue dot is fathe away fom the cente, so the ac it tavels is longe. MTH Wkst

6 Examples Hee ae some poblems fo you to ty and pactice some of the topics coveed in this section. 1. A 16 thin cust pizza is 16 in diamte and typically cut into 10 slices. (a) About how much cust does each slice have? Hint: Daw a pictue! (b) About how much aea does each slice cove? 2. The fist Feis Wheel built had 36 cas, was 264 ft tall, and took 20 minutes to complete two evolutions (complete otations). (a) How quickly does the Feis wheel otate? (b) How fast was each ca going? MTH Wkst

Math Section 4.2 Radians, Arc Length, and Area of a Sector

Math Section 4.2 Radians, Arc Length, and Area of a Sector Math 1330 - Section 4. Radians, Ac Length, and Aea of a Secto The wod tigonomety comes fom two Geek oots, tigonon, meaning having thee sides, and mete, meaning measue. We have aleady defined the six basic