WEBVTT
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In this video, we look at some applications of radian measure. We'll start with the
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area of a sector of a circle. So a sector of a circle is like a pie piece where the point
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is at the center of the circle and the two sides here are both radii.
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We're going to start by looking at the fraction of the circle represented by that sector.
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We could calculate this two different ways. If we knew the area of the sector,
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we can write that over the area of the entire circle, and that's going to tell us
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the fraction of the circle that the sector represents. We could do this another way.
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We could look at angles. So we're going to call this central angle of the sector theta,
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and if we look at the angle that would be if we went all the way around the circle,
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then that's one complete revolution or 2 pi radians. Either way we calculate the
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fraction of the circle present. It's the same fraction of the circle so these
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two ratios are equal. Well, we know a little bit more than what I've written.
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The area of this sector is what we'd like to find. Let's just call that A.
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The area of the entire circle, we can calculate that using an old formula from geometry.
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It's pi times the radius squared. Our radius is r. The central angle, theta, and then 2 pi.
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One thing I should point out is it's pretty important theta is in radians,
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since I've used 2 pi radians to represent the entire circle, all the way around it.
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Ok, so let's solve for A, the area of the sector. Multiply both sides by pi r squared.
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We can divide out those pi's and then sort of rearrange what's left.
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The area of the sector is 1/2 the radius squared times theta, with theta in radians.
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Now let's practice using this formula.
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In this example, we're asked to find the area of a sector of a circle where the
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central angle is 40 degrees and the radius is 5 centimeters. So we've just found
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this great formula. The area is 1/2 r squared theta, but theta needs to be in radians.
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So that's our first problem. So, whenever we're using a formula, we should always
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write it down before we begin, and then we can substitute in there. But as we
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discussed, this needs to be in radians. So we're going to take our 40 degrees
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and to the side here convert that to radians. How do we convert?
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Pi radians is 180 degrees, so we're going to multiply this by pi over 180 degrees. And
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for now, we're just going to reduce that to lowest terms and keep it as an exact value,
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so I can go a little bit further. That's really 2 pi over 9.
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Now we're ready to substitute. The area is 1/2 the radius, which is 5, squared
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times 2 pi over 9. I should stop here because my instructions didn't really tell me
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how to leave my answer. So we're going to start by giving an exact answer
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that's left in terms of pi, and then we'll approximate it if we need to later.
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Alright, so what can we do? Well 1/2 times 2 is 1, so I can write this as 25 pi over 9.
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What kind of units would the area of the sector have?
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The radius is given in centimeters, so area should be in square centimeters.
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This is an exact version of our answer. Sometimes you're asked to approximate
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your answer to one or two decimal places. At that point you would take your
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calculator out and multiply and divide on your calculator to get an approximation.
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Let's do that real quick.
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This is approximately 8.73. So we're going to use 2 decimal places here,
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centimeters squared. The number of decimal places will always be told to you
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in your homework or in any problem that you're working.
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Here's a second example of the area of a sector of a circle. Find the radius
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of a sector of a circle given the central angle is 3/5 radians and the area of the
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sector is 600 square feet. Round our answer to one decimal place.
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So we know we can use that formula. Area is 1/2 r squared theta for a sector of a circle.
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Theta needs to be in radians, so let's check that first. Great, this time we have
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our angle given in radians. But what's the difference in this problem?
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I'm not looking for the area. I already know that's 600 square feet.
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I know the central angle, but I need to find the radius. So the algebra
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in solving this one is a little bit more involved. So let's start by
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substituting what we know. The area of the sector is 600. The radius
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is what I'd like to try to find, and theta is 3/5 radians.
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I'm going to simplify the right hand side by multiplying those fractions.
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That's 3/10 times r squared, and I need to get that r squared by itself.
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So let's multiply both sides by the reciprocal of 3/10. That would be 10/3.
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So we have 10/3 times 600 equals, 10/3 times 3/10 is 1, so this is just r squared.
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10 times 600 is 6000, divide that by 3, so we have 2000 equals r squared.
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Now that's not a perfect square, so I'm going to grab my calculator
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so we can approximate it. r is going to be the square root of 2000.
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It's certainly going to be a positive number because it's the radius of our circle.
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The square root of 2000 is about 44.7 rounded to one decimal place.
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What would the units be for our radius?
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Well the area of the sector was in square feet, so our radius should be in feet.
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Next we're going to look at arc length.
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This is the length of an arc or a portion of a circle.
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So I've drawn a circle with radius r. This is the center point. I have central angle
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theta again. And the arc is labelled s, and it's this portion of the circle
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intercepted by this central angle. I'd like to be able to calculate
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the length of that arc knowing just the central angle and the radius.
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I'd like to look again at the fraction of the circle represented by, in this case, the arc.
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So we're going to do the arc length over the circumference, or distance around
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the entire circle, and then we're going to look again at the angle.
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So the central angle for this arc is theta, and an angle that starts here and goes
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all the way around the circle is 2 pi radians. So again, we're going to have that
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stipulation that theta has to be in radians. Now we've written the
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fraction of the circle present two ways. But they're the same fraction of the same circle,
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so they have to be equal. For arclength, we're going to use the letter s.
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Circumference of a circle, do you remember that formula?
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2 pi times the radius of the circle. Let's solve for s. Multiply both sides by 2 pi r.
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Notice that the 2 pi's divide out for us, and we're left with a very simple formula,
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s equals r theta, theta in radians.
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Let's practice using our arc length formula. We'd like to find the exact length of an arc
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of a circle with radius 10 feet intercepted by central angle 330 degrees.
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So we have this nice formula, s equals r theta, but theta has to be in radians.
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We're given the central angle of 330 degrees, so before we do anything else, we need
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to go ahead and change this to radians. There are pi radians in 180 degrees,
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so we're going to multiply by pi over 180. So this one is just 33 pi over 18.
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Oh wait, I can go further. Those both have a common factor of 3,
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so this is really 11 pi over 6. Now we substitute. The radius is 10 feet. Theta is
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11 pi over 6. Think of this as 10 over 1 and reduce your fractions.
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So I'm going to just go ahead and multiply, 10 times 11 is 110 pi over 6,
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and then reduce that. What kind of units would arc length have?
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Well it's just a length, so whatever the unit is used for the radius we'll use for
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our arclength so 10 feet. We wanted an exact value, so we want to leave it in terms
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of pi and an improper fraction.
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Here's another example that uses our arc length formula. Find the central angle
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of a circle with radius 30 inches that intercepts an arc of length 10 inches.
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So we can use our arc length formula, s equals r theta, and in this case,
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we're looking for the central angle. That's what we don't know.
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Our radius is 30 inches, and the arc length is 10 inches. So an arc of length 10 inches,
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this is the s in our formula. r is 30. If we divide both sides by 30,
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we'll have theta by itself already. So theta is 10 over 30 or 1/3. 1/3 what?
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What are the units for our angle theta?
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Right. So theta, in order to use this formula, has to be in radians.
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So when we solve this, we're finding our angle theta in radians.
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If a particular problem asks you for that answer in degrees, then you know how to
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convert from radians to degrees, and you can do that at that time.
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So you need to learn both of these formulas for area of a sector and for arc length.
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Don't forget that your angle, theta, has to be in radians each time that you use them,
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and then practice these examples until you're comfortable with these skills.