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## Perimeter of circular figures

# Angles, arc lengths, and trig functions — Harder example

## Video transcript

- [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.