Angles, arc lengths, and trig functions — Harder example

- [Instructor] In the xy-plane above, O is the center of the circle right over here. And the measure of angle AOB is five PI over six radians. If the radius of the circle is six, what is the y-coordinate of point A? Pause this video and see if you can figure this out before we work through this together. All right, now let's work through this. And if all of this seems really unfamiliar I encourage you to review degrees and radians or circle trigonometry on Khan Academy but I'll assume you have some familiarity with it. So first of all, they're telling us that the measure of angle AOB is five PI over six radians. So AOB, we're talking about this angle right over here, is five PI over six radians. We know that the radius of the circle is, so let me just use a different color, the radius of the circle is six. So we know that this distance right over here is six. We also know that this distance right over here, this is also radius, that is equal to six and they want us to know, they want to figure out what is the y-coordinate of point A? Well, the y-coordinate of point A which would be right there. We could also figure it out by figuring out, well what is this distance right over here that I am drawing in red, and this would be a right triangle. So let's think about, can we figure out what this angle is going to be? So you might already be familiar that when you're thinking in radians, two PI radians would go all the way around the circle and PI radians would get you halfway around the circle. So this angle over here is going to be halfway around the circle, which is PI radians minus the five PI over six radians, minus five PI over six. Now PI radians we can rewrite as six PI over six. So when you do the subtraction you are going to be left with six PI over six minus five PI over six is going to be equal to PI over six radians. Now this still might not be familiar to you. What is PI over six radians? Well, you could think about converting that to degrees. We know that PI radians is equal to 180 degrees because PI radians is halfway around the circle. So you divide that by six. This is equivalent to, let me write it this way. This is equivalent to 30 degrees. So if I write 30 degrees here, is a bell starting to ring in your head? Well, you might recognize this as a 30, 60, 90 triangle. How did I know that 60? Well, because if one side has a 90 degree, if one angle is 90 degrees, the other one's 30 degrees they all have to add up to 180. And this is a typical triangle you'll see a lot in your geometric career. So it's good to know about 30, 60, 90 triangles. And we also know that in 30, 60, 90 triangles the side that is opposite the 30 degrees is one half the radius. And that by itself lets us know what's going on because this is one half the radius that's what we need to figure out. The radius is six. So one half times six is equal to three and we're done. That is the y-coordinate over here. It is three.