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### Course: Class 10 (Old) > Unit 11

Lesson 1: Perimeter of circular figures# Angles, arc lengths, and trig functions — Harder example

Watch Sal work through a harder Angles, arc lengths, and trig functions problem.

## Want to join the conversation?

- how did sal know that the opposite side of the 30 degree angle was 1/2 of the radius?(17 votes)
- The 30 degree angle forms a 30-60-90 right triangle, which has special ratios between its side lengths. If the side opposite from 30 is x, the side opposite 60 is x*sqrt(3) and the side opposite 90 (hypotenuse) is 2x. We want to find the sine of the 30 degree angle. The sine of an angle tells us the ratio of the opposite to the hypotenuse in its associated right triangle. We know that the opposite side is 1/2 of the hypotenuse since the opposite side is x and hypotenuse is 2x, so sin(30) = 1/2.(14 votes)

- which calculator is more recommended scientific or graphic?

for SAT 1(9 votes)- All the questions on the SAT can be solved using a scientific or graphing calculator. There are no such question on the SAT 1 which can be only solved using a graphing calculator. Still, if you want you can take a graphing calculator to the test if you think that it will help you score more.(12 votes)

- why do you multiply 260/360 and 10pi(4 votes)
- 10pi is the entire circumference. 260/360 is the fraction that represents the segment of the arc, because the angle is 260 degrees out of a total 360 degrees. Basically, x is 260/360ths of the entire circumference, so you multiply the fraction by the whole to find the length of the arc, or the portion of the circumference. OR, there's a formula you use to find arc length. It's
`code`

where x is the measure of the central angle and C is your circumference. Hope that helps!`(x/360) * C`

(9 votes)

- im appearing for the digital SAT on 11th March. are the foundation vids of every topic enough prep?(3 votes)
- Hi!

I'm appearing for DSAT on 11th March too. It's important to practice as many questions as possible, and not just watch videos. That includes completing the short tests.

Also, the practice tests on the Bluebook app are really helpful.

Good luck!(7 votes)

- x=rcosθ

y=rsinθ

you know both r and θ

and done!(5 votes) - How do I solve equations along the line of, "What is the value of tan(x degrees)"(3 votes)
- convert degress into radians, and then evaualte tan, or you can use sin/cos(1 vote)

- I don't understand ! help(3 votes)
- What exactly dont you undertand?(1 vote)

- is there some way you can do this more efficiently using tan or tan^-1(2 votes)
- For the harder example video solution, can I do y coordinates equals r.sin(150)=3. I just want to know whether my method is correct because it gives me the same answer as the alternative method shown in the video but only less confusing.(2 votes)
- Is that a general rule to remember, the side being opposite of 30* equals 1/2 of the radius? (2:52)(2 votes)
- yes one half of the hypothenuse which is here the radius :

it comes from semi equilateral triangles if you have an equilateral triangle and you cut perpendicularly through on vertex you'll end up with two congruent right triangles having those three angles: 90,60,and 30. since you cut the equilateral triangle in half the new triangles have a side that is have the old side of the big triangle and another side which stays the same thus the small side would be have the other side which is a hypothenuse in the new small triangles(1 vote)

## 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.