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Hi everybody this is Mr. Bourne from Minnesota and his trusty cat and this tutorial will be on the unit circle radians angles and points now if you want to learn this fast this is the tutorial to watch what you will learn what the unit circle is how degrees and radians relate finding Radian measure and finding the XY exact coordinates of important Radian measures all under eight minutes first let's talk about what a unit circle is what you're seeing is an XY coordinate grid this is a plane with the pairs of coordinates and a unit circle is centered at the origin there it is little unit circle with a radius of one let's zoom in to get a little of a closer look at it here okay there we go now radius of one means that I could lift out this little blue piece here and this is the radius and as you can see no matter where I hold if it's got a radius of one exactly now if we take that radius, and we curve it here's a curved piece it's the same thing it's curved, and I've colored it purple if we take this curved piece, and we get a few more one two three more of them these will fit exactly the circumference of the circle but in just a second you'll see hang on that there is just a tiny little piece left over, so the radius will fit around the top hemisphere of the circle three times plus a little extra and this little extra which is right here is 0.14 long, so it takes 3.14 radii or radians to fit around the top half of the circle okay so let's move that little 0.14 piece in their okay 3.14 that is high so around the very top half of the circle is pi radians and the bottom half the circle is another pi radians, so the complete circumference around a unit circle is 2 pi radians this should be familiar if we go to the formula for the circumference of a circle its circumference equals 2 PI R well since we've got a radius of one that makes the math simple right there you go, so now you know how the formula got to be that way okay now I'm going to tell you how radians and degrees relate what you're seeing here is a slightly different picture of the unit circle and the red side which is moving around is going to be called the terminal side and the green side is going to be called the initial side, so I put it in your heads that a Radian is a length, and I'm going to adjust this so that there we go the terminal side is one Radian away from the initial side, so this is a length along the outer edge of the circle here, but a Radian can also be describing the angle formed by the terminal side and the initial side right inside here ok well let's bring degrees into this so there we go we see this is 57 degrees and that is the same thing as one Radian all right well it might not seem very natural, and they don't quite go together but let's just move this around here, and we'll see where the logical matchups are right here at exactly halfway point on the way to one complete pi Radian this is a PI over 2 radians and that's 90 degrees and over here we've got 1/4 PI or...
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