now the angle can be measured in degrees or radians so you need to be familiar with both so if you see a little circle on top then you know the angle is in degrees if you don't see that Circle then the angle is in radians for example if Theta is equal to 2 then you know that is in radians so this is two radians so anytime you see a number without this little circle its radians now what is a radian exactly what's the definition of a radian a radian is basically the measure of the central angle of a circle where the arc length is equal to the radius of the circle so let's draw a circle okay that is a terrible Circle let's do that one more time so let's say this is the center of the circle the radius is a distance between the center of the circle and any point on the circle now let's say if we draw the length of the radius but as an arc let's say that distance is also R if we connect the center to the edge of that point then that's also the radius so whenever you have a situation like this where the AR L is equal to R and is the same as the radius of the circle then the angle that forms is one radian so this angle is one radian so that's the definition of a radian a radian is the angle that occurs whenever the Arc Length which is basically this distance here is equal to the radius in fact if you want to find the angle Theta you can divide the Arc Length which is known as s by the radius and that will give you the angle in radians so keep this in mind let's say this is r and this is r and here's s The Arc Length so if you need to find the angle Theta simply divide the Arc Length by the radius so let's say if we have a circle and let's say the arc length is 12 CM so that's the distance between these two points let's call it point a and point B and let's say Point C is the center of the circle now let's say that the radius of the circle is 6 cm so that's the distance between the center and a point on the circle what is is the angle Theta in radians what is the angle so to find the angle it's the Arc Length divid by the radius so it's 12 CM / 6 cm and so this will give you two or two radians personally I like to think of the radius as being the length per unit radian I think of it as being 6 cm per radian and that way you can see how the unit ctim will cancel and in the end you should get radians so that's how you could find the angle let's work on another example so let's say the radius is 4 in and let's say the Arc Length s let's say s is equal to 16 in calculate the angle Theta as indicated in a drawing so the angle Theta is going to be S / R so in this case it's 16 in / 4 in per radian so this will give you 4 radians so now you know how to calculate the angle in radians if you're given the arc length and the radius now let's talk about how to convert angles from degrees to radians so what you need to know is that let's say if you have a circle for revolution of a circle is equal to 360° and that's equal to 2 pi radians now if 360 degrees is equal to 2 pi radians and if we divide both sides by two we can see that 180° is equal to Pi radians and this is the conversion factor that you want to use to convert from degrees to radians so let's say if we have the angle 60° and we want to convert it to radians here's what we need to do take your angle and multiply it by Pi / 180 you want the unit degrees to cancel so what's 60 / 80 for 1 we can get rid of a zero so becomes 6 over 18 so we have 6 piun over 18 but we can write 18 as 6 * 3 and we can cancel a six so therefore the final answer is pi over 3 so 60° is equal to pi over 3 radians let's try another example convert 150° into radians feel free to pause the video and try this problem so just like before we're going to multiply by Pi / 180 so let's begin by getting rid of a zero so now we have 15 Pi / 18 15 is 5 * 3 18 is 6 * 3 so we can get rid of a three so therefore the final answer is 5 Pi / 6 try these two examples convert - 225° into radians and also -30 so I'm going to start with -30 let's multiply that by Pi / 180 so once again we can get rid of a zero so this is going to be -3 pi and 18 I'm going to write it as 6 * 3 so we can cancel a three so the final answer is going to bek 6 that's equal to -30° now in this example let's multiply -225 de by Pi / 180 now it's helpful to know that 225 is 45 * 5 and 180 is 45 * 4 so now we can cancel 45 which will give us our final answer of -5 Pi / 4 so now you know how to convert angles from degrees to radians and that's all you need to do just multiply by pi over 180 now what about converting radians into degrees so let's say if we have 2 pi over 3 radians what should we do to convert this angle into degrees well this time instead of multiplying by pi over 180 we're going to multiply by 180 / Pi we need to flip it now you want to write it in such way that the pi letters cancel so this is going to be 2 * 180 / 3 now 180 is 60 * 3 so we can get rid of three and so we're left with 2 * 60 which is 120° and so that's the answer let's try some more examples try this one convert 11 Pi 6 into degrees so once again let's multiply by 180 / Pi and let's cancel Pi now we can write 180 as 30 * 6 so let's get rid of six so we just got to multiply 11 * 30 11 is basically 10 10 + 1 and if we distribute 30 to it 30 * 10 is 300 and 30 * 1 is 30 so our final answer is 330° here are some examples that you can try convert -7 piun / 4 into degrees and also -3 Pi / 2 so I'm going to start with this one let's multiply by 180 / Pi 180 is the same as 90 * 2 so we can get rid of two and 3 * 90 is 270 so the final answer is -200 and 70° now what about -7 piun 4 well let's do the same thing let's multiply by 180 over pi and cancel Pi now keep in mind 180 is 45 * 4 so we can cancel 4 so now we need to multiply 45 * 7 so that's the same as 7 * 40 + 5 now 7 * 40 if 7 * 4 is 28 7 * 40 is 280 7 * 5 is 35 if we add 280 and 35 this is going to be 315 so the final answer is - 315° now for those of you who want access to my complete online trigonometry course here's where you could find it uh go to emi.com and then in the search box you could just search for trigonometry and you can see my course is basically the one with the black uh background and then here is it I'm still adding more lectures but here's what I have so far um introduction into angles drawn angles converting degrees into radians uh linear speed angular speed problems Arc Length uh information on the unit circle how to evaluate trigonometric functions using the unit circle a right triangle trigonometry things like so TOA even you could have video quizzes as well solving work problems like angle of elevation problems and then you have the next section graph in s cosine functions secant tangent inverse trig functions pretty much all the common stuff that you'll see in a typical uh trigonometry of course even solving uh barings verifying trigonometric identities summon difference formulas double angle half angle and some other things too and as I mentioned before I'm going to add some other things as well so feel free to check it out when you get a chance and uh let's continue back to the video now you need to be able to draw angles in standard positions if it's given to you in radians we've covered degrees already so let's say if we have the angle pi over 4 how can we graph it and stand in position well for one thing it might be helpful to convert this into degrees 180 ID 4 is 45° and you know 45 is between 0 and 90 so we're going to draw the ray right in the middle between those two so therefore this is how you graph the angle pi over 4 let's try another example 7 Pi 6 where should we graph it well let's begin by converting it into degrees 180 is 30 * 60 and 7 * 30 is 210 now 210 is between 180 and 270 but it's closer to 180 so it's right there therefore this angle is 7 Pi / 6 try this one pi/ 6 so let's convert it to degrees first 180 ID 6 is 30 so this is zero and this is 90 so -30 should be in this region in Quadrant 4 so this is30 degrees now let's talk about some common angles that you need to know in radians and their locations let's start with pi over 4 so pi over 4 is located in quadrant 1 it's between 0 and 90 the next one that's similar to it they you need to know is 3 pi over 4 3 pi over 4 is 135 pi over 4 is 45 so 45 * 3 is 135 in quadrant 3 the next angle is 5 pi over 4 which is 225 and Quadrant 4 has the angle 7 pi4 7 * 45 is 315 now this angle here is 2 pi over 4 2 pi over 4 it reduces to Pi / 2 and whenever you see Pi remember Pi is equal to 180 so 180 ID 4 is 45 so that's how you can remember pi over 4 is 45 3 pi over 4 45 * 3 that's 135 5 piun 4 5 * 45 that's 225 which is in quadrant five now the next one 4 pi over 4 reduces to Pi and Pi is 180 here we have 6 pi over 4 which reduces to 3 pi over 2 so you won't hear 2 piun 4 6 Pi 4 4 Pi 4 because they're reducible fractions just know where they're located and that the fact that they're going to reduce to pi/ 2 3 pi over 2 or Pi but these are the main ones you need no Pi 4 3 Pi 4 5 piun 4 and 7 piun 4 now the next angles you need to be familiar with is pi over 3 so pi over 3 is about 60° so this is uh 1 pi over 3 and then 2 pi over 3 is in quadrant 2 so that's about 120 so remember Pi is is 180 so < / 3 180 / 3 that correlates to 60 so 2 piun over 3 is 60 * 2 which is 120 3 pi/ 3 simplifies to Pi which is 180 so you won't see 3 pi over 3 it just be Pi next you have 4 pi over 3 which is 4 * 60 that's 240 and then 5 piun over 3 that's 5 * 60 which is 300 so in terms of pi over 3es those are the common ones you'll see 1 pi/ 3 2 pi over 3 4 pi over 3 and 5 pi over 3 next we have pi over six so 1 pi over 6 which is 180 / 6 that's 30 and then we have 5 piun over 6 that's 5 * 30 which is 150 and then 7 piun over 6 that's in quadrant 3 that's 7 * 30 which is 210 and finally 11 pi/ 6 which is 11 * 30 or 330 and don't forget the angles that are on on the X and Y AIS this is 0° Pi / 2 which is 180 / 2 that's 90° Pi is 180 3 Pi / 2 that's 3 * piun / 2 which is 3 * 90 that's 270 so now you know where the common angles are located and so you can easily place them in their appropriate location if you understand uh everything that we just went over so those are the common angles that you'll see later in the unit circle so just make sure you remember those positions