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Course: Digital SAT Math > Unit 9
Lesson 5: Unit circle trigonometry: mediumAngles, arc lengths, and trig functions — Basic example
Watch Sal work through a basic Angles, arc lengths, and trig functions problem.
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- At2:37, why do you have to multiply by 6(pi)..? You already have the 360 degrees (which is the circumference of the circle in degrees), why do you have to multiply again by the circumference (in radians)?(16 votes)
- What Sal is doing is taking the entire circumference (6 pi) and multiplying that by the fraction of the circle represented by the arc ABC (135/360). One point to clear up: radians and degrees are measures of angles, not length. Radians are wonderful because we can convert them directly to unit length, but we can't do the same thing with degrees. Here's a helpful video on the topic: https://www.khanacademy.org/math/trigonometry/unit-circle-trig-func/radians_tutorial/e/radians-on-the-unit-circle(15 votes)
- how is the weather today?(14 votes)
- just use length of arc formula lol, its easier and faster(10 votes)
- Length of arc = theta*radius
Make sure your theta is in Radians.(6 votes)
- Isn't the formula for arc length , radius x angle subtended to it? I thought the answer was 405.(11 votes)
- radius x angle subtended is the formula to measure the arc length for the angles measured in radians whereas the angle in the question is in degrees. so the equation is 135/360 x 2pi x radius which is 3
so the answer comes to 9pi/4(1 vote)
- Can't we simply use the formula arc lenght=radius * thetha?(10 votes)
- Yes you can. Here is how you'd do that for this problem, referencing an answer by Alex.
arc length = r * θ = (3 in) * (135°) = (3 in) * (3π / 4 rad) = 9π / 4 in.(4 votes)
- Will this formula be given to you on the test? Or do you have to memorize it?(4 votes)
- Well, not exactly. You are given a couple pieces of it that you have to put together from your knowledge of arc length.
The formula sheet has formulas for area andcircumference
, plus the statement:The number of degrees of arc in a circle is 360.
If you don't remember how to do these, it would be great to practice them, plus the closely allied questions about area of a sector of the circle, given the central angle and radius.
The key to both kinds of questions is proportions. In other words, when you find the arc length, given the central angle, you are finding the fractional part of the whole circle circumference based on the whole 360 degree circle:
length of arc/length of circumference = central angle of arc/ total angle in circle
length of arc/C = central angle/360
m arc ABC/6π = 135/360
m arc ABC = (135/360) ∙ 6π
m arc ABC = (135/60) ∙ π
m arc ABC = 9/4 ∙ π or 9 π/4(11 votes)
- if the number of radians of arcs in a circle is 2(pi) how is a sector of a circle also 2(pi)?(8 votes)
- if pi is not pie? what is it?(2 votes)
- here pi means 180 degree(2 votes)
- its like follows
in order to find length of arc ACB
first angle subtended by it at centre would be 2pi - 2 pi/ 3 = 4pi/3
then just apply formula for length of arc
gives uc as 2 pi(6 votes) - How can 135/360 = 9/4? Really it is simplified down to 3/8. I'm so confused.(4 votes)
- No, he was using the following:
(135/160)x6
He left out the pi because it is in the answer so there is no need to work it out.
Hence (135x6)=810
So he is working with 810/360
Hence, he got 3/4
Hope this helps(3 votes)
Video transcript
- [Instructor] In the figure
at left, I pasted it up here, point O, point O is the center
of a circle of radius 1.5, we see that right over there. The missing sector of the
circle has central an, the missing sector of the
circle has central angle AOB equal to two pi over three radians, that's this right over here, and that's the central
angle for the missing, for the missing sector right over there. What is the length of arc ACB? Now ACB, ACB is this arc, it's kind of the rest of the circle, the part that's not missing,
so this is ACB right over here. Well the way that I would tackle this, I would think well
what's the circumference of the entire circle,
and then what fraction of the entire circle is this arc length? So let's first think
about the circumference of the entire circle. Circumference is equal to two pi times r. In this case, our r, our radius is 1.5. It's gonna be two pi times 1.5. Now two times 1.5 is three, and so this is going to be equal to, this is going to be equal to three pi. So the circumference of the
entire circle is three pi. And now there's a couple of
ways that you could do it. You could figure out
well what is the length of this arc right over here and then subtract that
from the circumference, and then you'd be left
with the magenta part. Or we could figure out the
central angle of the magenta part we could figure out this angle, and think about well what
fraction is that going to be if we were to go all the way around, and if we're thinking in radians, going all the way around
is two pi radians. So what fraction is this angle of two pi, and then that's going
to be the same fraction that this arc length is of
the entire circumference. Well what's this angle going to be? Well it's going to be, if we want, remember if we went all the way around, if we went all the way around the circle, if we went all the way around the circle, that'd be two pi radians. But if we wanna figure out
this magenta central angle, it's going to be two pi
minus this two pi over three. So this is going to be two pi, let me do it in that magenta color, so the central angle for
this piece of the circle that's kind of the central angle for ACB is going to be two pi
minus two pi over three. I'm going all the way around but then I'm subtracting out
this part right over here. Now what's two pi minus two pi over three? Let's see, I can find
a common denominator, instead of writing it as two pi, I can write that as six pi
over three, so let me do that. It's gonna be six pi over
three minus two pi over three. Well that's going to
be four pi over three, four pi over three. So once again this angle right over here is four pi over three. Four, four pi over three radians. Now what fraction is that of if we were to go all
the way around the circle? Well once again this central
angle is four pi over three, if you were to go all the
way around the circle, that's two pi, so this is the
fraction of the entire circle that this arc represents. And so let's just multiply that times the entire circumference, times three pi, and
let's try to simplify it. Let's see, we have pi divided by pi, and let's see if we take this three and multiply it times the numerator, this three is gonna
cancel with that three, and we're gonna be left
with four pi over two, four pi divided by two is equal to, is equal to two pi. So that's the length, that's
the length of this arc. It's actually exactly 2/3 of the entire, 2/3 of the entire circumference. So let me just select that,
two pi, and we're done.