Unit circle definition of trigonometric functions
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Unit circle definition of trig functions
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Example: Unit circle definition of sin and cos
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Example: Using the unit circle definition of trig functions
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Example: Trig function values using unit circle definition
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Example: The signs of sine and cosecant
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Unit Circle Manipulative
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Unit circle
Ferris Wheel Trig Problem Trigonometry problems dealing with the height of two people on a ferris wheen
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- All right, I have a problem here.
- Jacob and Emily ride a ferris wheel at a carnival
- in Billings, Montana.
- The wheel has a 16-meter diameter.
- So let me draw the wheel.
- It has a 16-meter diameter.
- So let me draw it big.
- Give me a lot of space.
- So it has a 16-meter diameter, so what's its
- radius going to be?
- Its radius is going to be half of that, right?
- So if I were to draw its radius, just draw it like that.
- It's a 16-meter diameter, so it's radius is going to be 8
- meters with its lowest point above the ground-- oh,
- with its lowest point 1 meter above the ground.
- So its lowest point is right here.
- This is its lowest point, and that is 1 meter
- above the ground.
- So this distance right here is 1 meter.
- Fair enough.
- Assume that Jacob and Emily's height h above the ground is a
- sinusoidal function of time where t equals 0 represents the
- lowest point on the wheel.
- So this is at point t equals 0 right here. t equals 0 is the
- lowest point of the wheel.
- Write an equation for h.
- Oh, I think I forgot, so let me reread it.
- Jacob and Emily ride a ferris wheel at a carnival
- in Billings, Montana.
- The wheel has a 16-meter diameter-- we did
- that-- and turns at 3 revolutions per minute.
- So it turns at 3 revolutions per minute with its lowest
- point 1 meter above the ground.
- Assume that Jacob and Emily's height h above the ground is
- a sinusoidal function of time, where t equals 0.
- So we need to write a function of h, their distance above the
- ground, as a function of time, and they're saying that
- time is given in seconds.
- So, first of all, they're telling us 3 revolutions
- every minute, right?
- So that's 3 revolutions per 60 seconds, and that's the
- same thing as 1 revolution per 20 seconds, right?
- I just divide both sides of the per by 3 for 20 seconds.
- And one revolution is how many degrees?
- One revolution is 360 degrees.
- So it's 360 degrees per 20 seconds.
- And if you're going 360 degrees per 20 seconds, let's divide--
- you know, the per you can just kind of use as an equal
- sign of the equation.
- 360 degrees for 20 seconds.
- That means you're going to go what?
- 18 degrees, Just divide both sides by 20.
- 18 degrees per second.
- And we could have done it with a numerator and a denominator.
- 3 revs per-- you know, you could have said 3
- revs over 60 seconds.
- That's actually how I should have done it.
- 3 revs over 60 seconds is equal to 1 rev over 20 seconds, which
- is equal to 360 degrees over 20 seconds, which is equal to 18
- degrees per second, right?
- So we're going to travel 18 degrees per second.
- So the total number of degrees we've traveled in t seconds is
- going to be-- so see, if we say the angle, that's the angle
- from our starting point.
- So let's say we've traveled t seconds, and we're right there.
- What is-- let's drop a little altitude right here.
- What is this angle going to be, where this angle is right here?
- What is this angle going to be?
- How many degrees have we traveled?
- Well, we say we traveled 18 degrees per second, so if we
- travel t seconds, this is going to be 18t degrees, right?
- All right, so let's see if we can figure out how their height
- as a function of this-- well, as a function of t or as
- a function of this angle right here.
- So what is this height right here?
- Up here?
- It's 1 meter plus the radius because this distance
- right here is 8.
- So we could say that this is-- this point right here is h is
- equal to 9 at this point, right?
- We could almost view that as the h axis.
- So that's h is equal to 9.
- So at this point, how high are they?
- If this is h-- so right now, let me draw a little
- drop and go flat here.
- So their height above the ground is this distance h,
- which is the same thing as this distance h.
- So what is that distance?
- Well, it's going to be-- well, if this distance is h, what is
- this distance going to be?
- This distance is going to be 9 minus h.
- How do I know this?
- This whole distance is 9.
- This distance is h, so-- let me do it in a better color-- so
- that this distance right here is 9 minus h.
- So let's see what we can do.
- What do we know?
- We know this distance.
- We know this angle is 18t degrees.
- And do we know this side?
- Sure.
- That's the radius.
- That's 8.
- 8 meters.
- 9 minus h meters, 8 meters, and 18 degrees.
- And what are these sides relative to this angle?
- Well, if we were to draw a triangle here relative to this
- angle right here, the 9 minus h is adjacent, and the 8 meters
- is, of course, the hypotenuse, right?
- So what trig function deals with adjacent and hypotenuse.
- SOHCAHTOA.
- CAH, cosine is adjacent over hypotenuse.
- So we could say the cosine at 18 degrees, the cosine of 18t
- degrees, is equal to its adjacent side.
- The adjacent side is 9 minus h.
- It's equal to 9 minus h over the hypotenuse, over 8.
- And now we can solve for h, and we'll have h
- as a function of t.
- So we multiply both sides by 8.
- You get 8 cosine of 18t is equal to 9 minus h.
- Maybe we could subtract 9 from both sides.
- So we get minus 9 plus 8 cosine of 18t is equal to minus h, and
- then multiply both sides by negative 1, and then you get
- 9-- positive 9, right-- minus 8 cosine of 18t is equals
- to h, or h is equal to 9 minus 8 cosine of 18t.
- So there we have it.
- We have expressed h as a function of t.
- And in the next video, I'm actually going to
- graph this function.
- See you soon.
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At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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