welcome back mathematicians in this video we are going to discuss radians radians represent another way of measuring an angle so at this point your math career you have likely used degrees to represent angles we are now going to introduce radians as an alternative way to measure angles the radium measure theta of an angle is the measure of the ratio of length of the arc it spans on the circle to the length of the radius and a more concise way to write it you have theta which is the angle in radians is equal to the arc length over the radius and we let the variable s represent the arc length and we let the letter or variable r represent the radius to give you a diagram of what this means if we were to put an angle in standard form where the initial side is the pi on the positive x-axis and the terminal side let's go with in quadrant number one s would represent the length of the arc from the intersection point of the initial side to the intersection point of the terminal side so this would represent the distance s or the arc length r represents the distance from the center of the circle to any point on the circle so both of these distances would be considered r from the center of the circle to the point where the the rays intersect the circle and theta would be the angle from the initial side to the terminal side represented in radians we can then use the previous formula to come up with a new formula that allows us to calculate the arc length the arc length theorem says for a circle of radius r a central angle the central angle is a positive angle whose vertex is the center of the circle of theta radians subtends and subtends means the rays pass through the circle and arc whose length is s that formula which again remember the formula we just discussed was theta is equal to the arc length over r the radius if we solve that formula for s by multiplying both sides by r what we end up getting is s is equal to r times theta and again a ser a picture or a diagram would be if i have a circle and i have a central angle that subtends or intersects or passes through the circle we can then calculate this distance along the arc by taking r which is the radius times theta which is the angle and theta needs to be in terms of radians not degrees so for our problem here we have find the length of the arc of a circle of radius 4 meters subtended by a central angle of 0.5 radians so again we have a circle here we have a central angle we have the radius which is the distance from the center of the circle to the intersection point of the ray and the circle and this is going to be four meters and then we have the angle and the angle is going to be 0.5 radians so in order to calculate that arc length you're going to take s is equal to r times theta which is equal to 4 times 0.5 which is equal to 2. and because the rate because the radius is in terms of meters your arc length would also be in terms of meters we should now be able to convert between degrees and radians since both are ways to measure angles circumference or the measure of one complete revolution can be represented with the formula capital c is equal to two times pi times r one revolution around a unit circle and a unit circle is a circle with a radius of one is c is equal to two pi this means that one revolution around a unit circle is equal to two pi radians or 360 degrees because one full revolution can be represented as 2 pi radians or 360 degrees we can then set 2 pi radians equal to 360 degrees dividing dividing both sides by 2 allows us to get the formula pi radians is equal to 180 degrees so if we were wanting to convert from radians to degrees we would then say we were going to multiply the angle by 180 degrees over pi radians and that would be the identity ratio we would use if we wanted to convert from degrees to radians we would then use the identity ratio pi radians over 180 degrees we can now use that information in order to convert angles from degrees to radians so let's go ahead and multiply by the correct identity ratio in this case since we're converting from degrees to radians we would want to multiply by pi radians over 180 degrees this will then reduce by a common unit of degrees in both the numerator and denominator and from there we would want to go ahead and multiply across this will give us a product of 80 pi over 180 radians from there we would want to reduce the fraction to get a result of 4 pi over 9 radians so that means 80 degrees is equal to 4 pi over 9 radians let's now go to the second problem where we have 100 negative 135 degrees we're going to multiply by the same identity ratio so pi radians over 180 degrees and again we would reduce by the common unit in both the numerator and denominator of degrees then multiplying across we will get negative 135 pi over 180 radians from here we would go ahead and reduce and reducing we will get negative three pi over four radians now we're going to convert each of these angles that are in terms of radians 2 degrees once again we're going to multiply by an identity ratio this time we're going to multiply by 180 degrees over pi radians these angles are in terms of radians while they are not labeled as such and so we can reduce by the common unit of radians in both the numerator and denominator this then gives us a final answer in terms of degrees now we will go ahead and multiply across so i will also by the way reduce by a common factor of pi in both the numerator and denominator so again going back to the multiplication we're going to multiply across with the fraction so we'll get a product of negative 540 over 5 degrees and so then we reduce the fraction and what we end up getting is negative 108 degrees let's go to the second problem and so we will again multiply by the same identity ratio it's 180 degrees over pi radians and again the problem or the angle is in terms of radians so we'll reduce by the common factor of pi and also the common unit of radians leaving us with an answer in terms of degrees let's go ahead and multiply across and so when we multiply across we'll get 1440 over 3 degrees we then need to reduce this fraction and so what we end up getting is 480 degrees all right guys good luck