this is section 5.1 angles and their measure and we will be wanting to find degree measure first and in order to do that we will need to talk about an angle its initial and terminal side and of course the vertex so looking here uh to the right you can see we have the vertex right here and then we have the initial side which is the starting position and we rotate from that initial side to the terminal side now an angle is in standard position if its vertex is at the origin in the x y plane okay so as you can see if we have a vertex here then we have an angle in standard position and you can see over here to the left that the initial side is positive x-axis and here the vertex is at the origin and that is an angle in standard position now the measure of an angle can be positive or negative and if the angle rotates counterclockwise then the measure of the angle is positive and if the rotation is clockwise as you can see here we're going to have a negative angle and you can see this angle whoops this angle is negative 150 because we rotated counterclockwise here our angle is positive and a degree is one way to measure an angle now for a full rotation of a ray about its end point we would have to rotate 360 degrees that is a full rotation and now you could think of one degree is 1 360 60 of a full rotation let's go ahead and just take our time and begin by sketching angles in standard position we will actually sketch both of these as you can see 100 degrees so we're going to begin in standard position and we're going to rotate 120 degrees now if this entire plane is 360 then 360 divided by 4 so this would be 90 degrees now we need another 30 degrees so we will actually continue on to here and we could just go ahead and draw that and we would write 120 degrees and what do you notice we notice that we are in quadrant two once we have this measure of this positive angle and we rotate 120 degrees counterclockwise we're in quadrant two let's now we have negative 70 degrees so we will be rotating clockwise again this is 90 degrees so we're not going to go quite to 90. so we'll go just before 90 maybe about right there and then we will write negative 70 degrees and what do you notice we are in quadrant four okay now we're going to have some very special angles and because they're so special they have names so we'll look at a right angle and that measures 90 degrees and as you can see that's one quarter turn then we'll have a straight angle and as you can see we rotate one half of a turn and we have gone counterclockwise from here to here and then acute an acute angle is an angle that is less than 90 degrees and an obtuse angle is greater than 90 degrees and we say 90 to 180 so as you can see both of these angles below note acute and obtuse remember this is 90 degrees so that is why this angle is acute because it is less and since this is 90 degrees and we rotated more than 90 that's why we have an obtuse angle okay we also have some special angle names of complementary angles and supplementary and this deals with two angles so these are all names for single angles and these are properties with two angles these are two angle properties okay and so if you have a complementary if you have complementary angles if you add up the measures of the angles they will equal 90. so what would you say that the complement of a 30 degree angle would be that's correct 60 degrees why is that because 30 plus 60 equals 90. so these are complementary angles how about supplementary this is when the sum of the two angles is 180 degrees so the supplement of 130 degree angle must be a 50 degree angle that's correct and that is because 130 plus 50 is 180. okay now let's get a little more detailed about degrees a degree can be divided into 60 equal parts and these are called minutes that's kind of easy to remember and then each minute is divided into 60 equal parts called seconds so we have 60 seconds in a minute and we have 60 minutes in a degree and you can see those right here so let's do some converting let's convert to decimal degrees so we're given 38 degrees and 12 minutes so how would we convert that well let's start with 38 degrees and we have 12 minutes added to that now we're going to have to convert minutes to degrees so as you can see here we have one degree is equal to 60 minutes right here so whoops so we will have and i don't have to put that parenthesis there i can just put it times times 1 degree is 60 minutes now these minutes will cancel and we're just left with degrees so we know that we're doing pretty good there with our units okay so 12 divided by 60 is one-fifth so 12 divided by 60 is one-fifth so we will cancel these out and put one fifth okay so now what do we have this is equal to 38 degrees plus and what is one-fifth and i can go ahead and write that here one-fifth of a degree and one fifth is point two so we have 38 degrees plus point two degrees see if i can make this a tad bit smaller i made that awful large and now we have moved this over and now we have 38.2 degrees now let's go the other way let's convert to degrees minutes and seconds and we will begin with 120 0.45 degrees now let's convert this .45 degrees and we will have 120 plus 0.45 degrees and we will multiply that by 60 minutes divided by 1 degree so there are 60 minutes in one degree so now we have .45 times 60 you can do that in your calculator if you want so we have 120 plus .45 times 60 comes out to be 27 and again these degrees will cancel and we just have minutes left so now we have 120 degrees and 27 minutes okay so both of these conversions depended on what one degree equals 60 minutes that was very handy for those two examples now we're going to move on to radian measures and radian measures are different than degree measures and you're going to have to know the difference and when you're using your calculator you will have to use either radian mode or degree mode so now let's go ahead and take a look at what we have when two lines or rays cross a circle the part of the circle between the inner section points is called the intercepted arc and it's often denoted by s and you can see that s right here a central angle is an angle with the vertex at the center of the circle so you can see the central angle there shown so a central angle that intercepts an arc on the circle with length equal to the radius of the circle has a measure of one radian now that may seem like a lot but i think as we work through these examples you will see that radians tend to be quite easy to work with so one radian can be written as one we have no units on radians they say it is unit less okay so now let's talk about a radian measure of an angle the radian measure of a central angle theta subtended by an arc of length s so the arc length is s and the central angle is theta we can now write that theta equals s over r and so that is the radian measure of an angle now the number of radians in one revolution if we would write s over r this would be the entire circumference of a circle which you know already is 2 pi r divided by the radius and what happens to those r's they cancel and you're just left with 2 pi and 2 pi is about 6.28 so the measure of one full revolution in degrees is 360 degrees so what do we know now we know that 2 pi equals 360 degrees if you'd like to divide both of those by 2 then you will have pi equals 108 degrees either way okay so how are we going to convert between degree and radians just with what we have here you can use pi over 180 degrees if you want to convert degree to radians or if you want to convert from radians degrees starting with radians you'll multiply by 180 degrees over pi make sure you're using your units it makes everything a whole lot easier let's begin with converting from degrees to radians as you can see we're starting with degrees and we want to end up with unitless radians okay so how are we going to do that we'll begin with 120 degrees so we want degrees in the denominator to cancel that and as we said there are 180 degrees in pi radians and if we take 120 and divide it by 180 you will see that you will get two thirds let me show you that in the calculator i tend to let students follow along in the video with their calculator and pause but i just want to make sure you know that if you have 120 divided by 180 you can press this math button and then press frac number one and press enter and it will give you the fraction which is two-thirds so i will assume you will be working on your calculator alongside and checking those uh reductions when i give them to you so now do we what do we have left we have a two and a pi left i didn't write that very neat so we have a two and a pi left divided by three okay so now well that's not that didn't do very good job there okay so now how about negative 300 degrees we're going to use that same factor pi divided by 180 degrees because we want these degrees to cancel so that we're left with radians so now when we divide 300 by 180 you can try that in your calculator or negative 300 divided by 180 you will be left with a negative five over three and so now we have negative five pi over 3 radians now let's go the other way we're going to convert from radians to degree and we will just reciprocate what we just used so now we're going to multiply this by 180 degrees equals pi or that same factor so as you can see here the pies whoops the pies are going to cancel and we'll have 180 divided by nine so 180 divided by nine that will be 20 over one so we can't forget about that 8 in that numerator as you can see so we end up with 8 times 20 which is 160 degrees okay so now let's try one more of those negative 1.8 so we'll multiply again by 180 degrees over pi and now as you can see we will just multiply right across and we will end up with a negative 1.8 times 180 is a negative 320 4 degrees over pi now if you put that in your calculator i'll show you how to do that so if we go ahead and put a negative 320 and we divide that and as you can see pi is above this arrow but we need to have second arrow up and there we will get an approximation i don't know why that did that let me try that the opposite of 324 divided by pi and we'd get negative 1 i think i used a minus sign a negative 103.132 let's just go to two decimal places so that was 10313 so negative 103.13 degrees okay now let's talk about co terminal angles okay and coterminal angles are two angles in standard position with the same initial side and the same terminal side so let's take a look these angles in standard position are coterminal if their measures differ by a multiple of 360 degree degrees or multiple of two pi so in other words we will just be looking to see if they had an additional rotation so taking a look at this first example if we have theta equal to 50 degrees and theta equal to 410 degrees and we add those together we will get uh if we add 360 degrees to 50 we will arrive at 410 degrees and as you can see we rotated 50 and then we rotated another 360 and ended up at 410 and i'll show that show you that so this is 50 degrees we rotated 360 degrees and we ended up back here again so we have coterminal angles these are co terminal angles because their degree measures differ by 360 degrees okay now let's look at this next example theta equals pi over 6 and theta equals negative 11 pi over 6 and as you can see if we subtract 2 pi remember we can add or subtract it's a multiple we come up with negative 11 pi over six so um and and doing that subtraction um you can do that by hand or let me just show you this real quick pi over 6 minus 2 pi is the same as 1 6 pi minus 2 pi and if on your calculator you take 1 6 and subtract 2 i'll go ahead and show that to you just in case so we have 1 6 minus 2 math frac will have a negative 11 6. so we have a negative 11 6 pi and that's the same as this a negative 11 pi over six okay so now let's just try a couple examples we're going to be looking for a positive angle and a negative angle that is coterminal to the given angle so we're looking for a co-terminal angle okay coterminal angles are the angles that differ by these multiples of 300 degrees or 2 pi radians so how are we going to do that well since we're given degrees we could take 110 degrees and add 360 degrees and we'll come up with 470 degrees now if we have 110 degrees and we subtract 360 degrees we'll have negative 250 degrees so 110 and negative 250 are coterminal angles and 110 and 470 are what we call co terminal angles okay now we're given radians here so we're going to have negative pi over 4 plus 2 pi and negative pi over 4 minus 2 pi we're going to do the same thing we did last time we have a negative 1 4 pi plus 2 pi and we have a negative 1 4 pi minus 2 pi and if we do those in our calculator if you have negative 1 4 plus 2 you will have 7 pi over 4 or 7 4 pi and again we will have a negative nine fourths whoops fourths pi and we really need to put that pi in the numerator uh i kind of split that out in order to solve those but we could put it like that okay okay how about number seven find an angle between zero and 360 that is coterminal to this angle 910 now keep in mind we have to continue to find the difference of 360 degrees until we get an angle between 0 and 360. so let's just start subtracting we have 910 minus 360 and we come up with 550 degrees let's do it again because 550 is not between 0 and 360. let's subtract again and we get 109 degrees and there we have it so what would we also say we could say 910 degrees minus 2 times 360 degrees is equal to 190 degrees okay i guess i could have put that right there okay let's try negative 80 degrees let's start adding 360 degrees to this angle so plus 360 degrees and um because we need to find an angle between 0 and 360. so we're going to have to go 4 we're going to have to move in a positive positive degrees 360 positive degrees so if you add this together you come up with 280 degrees so that makes it really nice let's go to radians find an angle between 0 and 2 pi that is coterminal to the given angle so here we go we need to find an angle between 0 and 2 pi and we're already at 17 pi over 6 so we're going to have to subtract 2 pi and see what we get so again if on the calculator whoops if you want to take 17 divided by six minus two we come up with five sixths so we will have five sixths pi okay now let's try b let's we need to and 5 6 is between 0 and 2 pi okay 5 6 is just under one so that's just under pi okay going to be negative 35 pi over 12 we need to try to find an angle between zero and two pi so we're going to have to start adding 2 pi and let's see what we get negative 35 divided by 12. and we're going to add 2 let's math frac that and we have a negative 11 pi over 12. so we are not we have a negative angle and we must find one between zero and two pi so we're going to have to do that again and we're going to have to now um find a negative 11 pi over 12 and let's add another two pi and so see what we get let's add two to that and now oops i might have i might have done um that uh incorrectly let me clear that so negative 11 divided by 12 plus 2 and we get 13 over 12 and we are between 0 and 2 pi if we are at 13 pi over 12. okay now let's end our lesson of course we'll have um actually we're not ending with this arc length we will uh talk about um linear and angular speed after this this is quite a big section and a lot to learn in one section okay so now let's talk about the arc length of a sector of a circle so if we're looking for the arc length and we're given a circle of radius r the length of this arc length right here is r times theta so it's this radius times theta that is the length of this arc so let's try one find the exact length of the arc so here we're finding the length of an arc intercepted by a central angle theta on a circle of radius r and we'll round to the nearest tenth of a unit okay so they give us theta and they give us r so we might as well find the arc length is r theta and that would be 14 times pi over 12. and if you take 14 and divide it by 12 you will have 7 over 6 and we end up with 7 pi over 6 meters okay since this is meters now let's try b we're given angle we're giving angle measure in degrees and r but we still use the same formula so the arc length is r theta and that's going to be 2 times 20 degrees and i really should put feet in there but i try to just wait on my units until i'm finished 2 times 20 well actually we need this in radians sorry about that because we have to have theta in radians only okay so let's go over here to the side of our paper and if we have theta move this theta is equal to 20 degrees and remember we're going to have pi divided by 180 degrees and that's equal to pi over 9 radians okay so those degrees canceled remember and then 20 over 180 is 1 9. okay i'll try not to write over those a lot so you can still see that but anyway so now we have two times pi over nine and we have two pi over 9 feet okay so this is 2 feet radians are unitless so that's why we have feet over here in part a we have pi over 12 theta is pi over 12 radians but we have 14 meters and that's why we end up with meters okay taking a look at number 10 key west florida is located at 24.1 degrees north 81.8 degrees west and cleveland ohio is located at 41.5 degrees north and 81.7 degrees west since the longitudes are nearly the same the cities are roughly due north and south of each other using the difference in the latitude approximate the distance between the cities assuming that the radius of the earth is 39 60 miles okay so how are we going to look at this let's go ahead and write key west down and that's 24.1 degrees north and we have 81.8 degrees west and then we have cleveland is 41.5 degrees north and 81.7 degrees west they are saying that these are nearly the same so they would just like for us to look at these latitudes okay so what are we going to do let's go ahead and subtract them so we have 41.5 degrees minus 24.1 degrees and we'll have 17.4 degrees now we need theta in radians okay so we have theta equal to 17.4 degrees and you guessed it we're going to have pi divided by 180 degrees again those degrees will cancel and if you put that in your calculator you'll have 0.3 and that's radians we won't write those units but i will put that in parentheses so now what can we do we can say that and they asked us to find the distance between the cities which is going to be that arc length and that's r times theta and that's equal to 39 60 miles and remember theta is in radians with no units and we have 1188 miles so that was kind of a cool problem and you'll have lots of practice with those and i think that makes makes sense okay let's now look at linear and angular speed so in order to compute linear and angular speed we need to find out what these definitions are and they're quite simple especially after you have used them so angular speed and linear speed are a little bit different so angular speed uses this symbol and linear speed we'll use this symbol whoops v okay so w and v but so what will we have we will give those a try so angular is theta divided by t which is the time and linear speed is this arc length divided by the time and of course we know that s equals r times theta and that's why we have this now what do you notice do you notice that if we have theta over t is the angular speed theta over t right here is the angular speed so that is where we come up with r times the angular speed for linear speed so it depends on what you're getting so let's look at number 11. a wheel with a diameter of 1 foot spins at a rate of 40 revolutions per minute what is the angular speed so let's go ahead and look at our angular speed we have w equals theta over t and again theta is that w is given as 40 and we have revolutions in one minute so they really gave us that angular speed they gave that to us but now we are going to convert that to radians okay so how do we do that we have 2 pi radians in one revolution and so when we work that out we come up with 80 pi radians per minute or since radians are unitless we have 80 pi per minute okay let's look at b how fast in feet per minute does a point on the edge of the wheel spin so now we are looking for linear speed okay so if we're on the edge of the um wheel that would be our linear speed okay so we have the equals r times w since we have w here so that radius we would use if the diameter is one if the diameter is one foot that means the radius is 0.5 feet or you could put a half okay so we will have 0.5 feet we'll use our w which is 80 pi per minute and working this out we come up with 80 times 0.5 or 80 times a half is 40 pi feet per minute and if you'd like you can take 40 and multiply it by pi in your calculator and you will get about 126 feet per minute okay last thing is we're going to be looking at the area of a sector of a circle and it is quite simple to uh to look at a sir uh the entire area of a circle and then just find the area of this sector now how would we do that we will use theta which is this theta we will use the radius which is this radius and we will square it and then we will multiply by a half okay so let's try one of these a circle has a radius of 12 feet so let's go ahead and just write that is 12 there and actually if that is 12 this is also 12 those are the radii and we have 130 degrees so we passed up 90 and we're going there and we have 135 degrees okay and now i guess i didn't need this arrow there just go ahead and put 135 degrees so now they've asked us to find the area of that sector so we're going to use this equation so area is one half and now we need r squared and so that would be 12 feet squared but now we need theta in radians okay so working on that over here if we have theta equals 135 degrees remember we have pi over 180 degrees we're multiplying it by because one pi is equal to 180 degrees and you can reduce that this comes out two three over four and we end up with three pi over four so that's what we're going to put right over here three pi over four now we have to be careful because we're going to have a one-half a 12 and a three-fourths okay and that 12 has to be squared so why don't i write that out we have one-half times 12 squared times three fourths times pi and all of that is going to be feet squared so this is what you'll be doing in your calculator and so i hope that that helps you might even want to write this out times 144 times three-fourths and then we have that pi again and our units and if you multiply all of that out you'll come up with 54 pi feet squared or if you put 54 in your calculator and multiply it by pi you will have 169.6 feet squared and that concludes this section