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 (part 2) Part 2 of the ferris wheel problems. Graph of h(t)=9-8cos(18t)
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- In the last part of the problem we figured out that the
- function of the height of the ferris wheel, the people at the
- ferris wheel at any time is a function of t. h of t is equal
- to 9 minus 8 cosine of 18t where t is in seconds.
- Now the second part of this problem they want us to graph h
- as a function of t between 0 is less than or equal to t, is
- less than or equal to 30.
- So let me draw axes.
- So let's say that that's my h-axis.
- Let's say that this is my t-axis.
- So this is t equals 0, and this is t is equal to 30 seconds.
- I get confused when I see this 18 here or whatever.
- So what I'm going to do first of all is I'm going to graph a
- different function, slightly different function, then I'll
- translate it to this function.
- I'm going to graph h of theta is equal to 9
- minus 8 cosine of theta.
- I think you'll see where I'm going with this
- when I'm all done.
- So let's try to graph h of theta is equal to 9
- minus 8 cosine of theta.
- So when t is equal to 30 seconds, what
- is theta equal to?
- So 30 times 18, that's 540.
- So this is 540 degrees.
- Same thing.
- I'll write the thetas in red above the t-axis.
- This is 540 degrees, so that's like two times
- around the circle.
- So that's 540 degrees, then this is going to
- be roughly 270 degrees.
- So, 90 degrees will be about 1/3 of this.
- That would be 90 degrees, that would be 180, so that would be
- 90 degrees, that would be 180 degrees, this would be 360
- degrees, and this would be 360 plus 90 so this will
- be 450 degrees.
- If you wanted to figure out the corresponding time,
- you just take this degree and divide by 18.
- So it's 90 divided by 18 is what?
- It's five, right?
- So if I were to write here, this is at 5 seconds, this is
- at 10 seconds, this is 15 seconds, this is 20 seconds,
- this is -- sorry, this is 25 seconds, this is 30 seconds.
- Actually, a simple thing we could do is let's just figure
- out what the value of the function is at these points.
- Because these are pretty easy degrees to figure out
- what the cosine value is.
- So let's figure out -- let me draw a table.
- Tables are always good and I'll do it in yellow.
- So I'll draw a t theta and h.
- This might be kind of an unconventional way of doing
- things, but I have a simple mind so this is actually
- how I like to do it.
- So I like to think of theta as 0, 90, 180,
- 270, 360, 450 and 540.
- And t, the corresponding time of those, as 0,
- 5, 10, 15, 20, 25, 30.
- It's not rocket science here.
- When t equals 15 seconds, 15 times 18, we're trying
- to find the cosine of 270 degrees, right?
- 15 times 18 is 270 degrees.
- I'm just doing this because I don't have a calculator and
- this will help me pick good points.
- So when t is equal to 0, what is height?
- Or t is equal to 0, theta is equal to 0, so cosine of theta
- is -- cosine of 0 is 1.
- So 9 minus 8 is 1.
- I'm going to do h in a different color.
- So this is 1.
- Cosine of 90 degrees?
- Cosine of 90 degrees is 0.
- So 9 minus 0 is 9.
- Cosine of 180 degrees?
- So we're going all the way around the unit circle.
- Cosine of 180 degrees is minus 1.
- So minus 1 times minus 8 is plus 8, so 9 plus 8, that's 15.
- Cosine of 270 degrees are pointing straight down, so the
- x-coord is going to be 0.
- So once again, we're at 9 again.
- 9 minus 0.
- 360 degrees.
- Cosine of 360 degrees is the same thing as
- cosine of 0, right?
- So once again, I mean we've gone around the circle once.
- So it's going to be the same as 0, so it's going to be 1.
- And 450 is going to be the same thing as 90.
- So it's going to be 9 and then 15 degrees.
- So let's plot these points.
- Actually, let me just draw 15 up here.
- So what are the points that keep showing up?
- So this is 1, that's 1, that's 1, and then we have 9.
- 1, there's 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15.
- Fair enough.
- So let me draw some guidelines just to help us.
- Actually, let me do them kind of hard to see, because I
- don't want to draw too much attention to the guidelines.
- I could do one guideline there.
- Then I'll do a bottom guideline right there.
- Then the 9 keeps showing up.
- Oh, you know what, I can't add.
- What's 9 plus 8?
- It's not 15, it's 17.
- Sorry, clearly, I need to practice my addition.
- So this is 9 plus 8, this is 17.
- And I realized that because I was like, well 9 should be in
- the middle, so this is actually 17.
- Ignore my little marks here.
- That's 17, this is 1.
- Ignore the marks.
- 9 would be right in the middle between 1 and 17.
- So let me draw kind of mediant point right there.
- So this is 9.
- Sorry I can't add properly.
- Then let's draw the graph or at least plot the
- points on the graph.
- So, t equals 0 where h equals 1, so that's this point.
- That's right here.
- When t equals 5, h is equal to 9, right here.
- When t is equal to 10, h is equal to 17.
- When t is equal to 15, h is 9 again.
- So it's right here.
- At 20 we're back at 1.
- I think you see the pattern.
- At 25 we're back at 9.
- And then at 30 we're back at 17, not 15, because now I
- have corrected my mistake.
- And this is going to be sined graph, it's going to look
- something like this.
- Let me do it in a vibrant color so I can overwrite everything
- and it's going to look something like this.
- Go oops, and then up and them down here.
- Curve up, come back down, curve up and then come back down.
- Like that.
- So that's our graph.
- I think in the problem they tell us to approximate.
- Actually, let me open up my cousin's problem -- my other
- account has timed-out on me while I recorded this.
- They wanted to approximate when t equals 4 what the height is.
- So when t equals 4 the height is like right
- around there, right?
- So the height is a little bit less than 9.
- And I don't know, 7 or 8 meters in the air.
- And when time is equal to 10 -- well, time equal 10, we figured
- out exactly, we know that there are 17 meters in the air.
- So I know this was kind of a little messy and graphing
- trick functions tend to be, but hopefully you found
- this vaguely useful.
- Have fun.
Be specific, and indicate a time in the video:
At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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