Graphs of trig functions
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Example: Graph, domain, and range of sine function
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Example: Graph of cosine
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Example: Intersection of sine and cosine
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Example: Amplitude and period
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Example: Amplitude and period transformations
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Example: Amplitude and period cosine transformations
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Example: Figure out the trig function
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Graphs of sine and cosine
Determining the equation of a trigonometric function Determining the amplitude and period of sine and cosine functions.
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- Hello.
- I'm now going to use the actual Khan Academy website to
- do some more problems.
- And this time we're going to go the other way around.
- We're going to look at the graph of a trigonometric
- function and we're going to try to figure out the equation.
- So let's start with the problem we have in front of us.
- So we have a-- well we don't know if it's a sine curve or a
- cosine curve, but I guess it's fair to say that it's
- one of the two.
- Actually, let's answer that question first.
- What do you think this is?
- Do you think that this is a sine curve or a cosine curve?
- What's the difference?
- Or what's the easiest difference to differentiate
- between the two?
- Well, what was sine of 0?
- Let me get a little chalkboard in here.
- Let me bring it right here.
- Let me make sure my pen tool is correct.
- All right.
- What is sine of 0?
- And it could be 0 degrees or 0 radians.
- Well if you remember from a couple of the other modules,
- or even if you want to use a calculator, sine of 0 is 0.
- And what is cosine of 0?
- Well cosine of 0, if you remember from the last modules
- or you want to use a calculator-- although you
- shouldn't have to use a calculator for cosine of 0--
- you might remember that it was 1.
- So this graph that we have here, it's definitely when x is
- 0, when the x-axis is 0, the function definitely isn't 0.
- In fact, in this case it's 1 1/2.
- So this tells us that this isn't a sine curve-- although
- later with shifts we'll learn that it could be a shifted sine
- curve-- that this isn't a sine curve, that this is
- a cosine curve.
- And then you might ask, well Sal if this is a cosine curve
- why is f of 0 equal to 1 1/2, or 3/2, instead of 1?
- Because I just said here that cosine of 0 is 1.
- Well that's because there must be some type of a coefficient
- here, let's call it A, that is changing the amplitude
- of this cosine curve.
- And if you remember from the last module, what
- do you think this A is?
- Well that A is just literally the amplitude of the curve.
- And what's the amplitude of this curve?
- Well, the amplitude of this curve, if we just see how much
- it moves above and below the x-axis, well it's that 3/2,
- or that 1 1/2, we've been talking about.
- See it moves up 3/2, and it moves down 3/2.
- So let me just write that.
- So we know that this is 3/2 cosine of, well, something x.
- Right?
- We know f of x is equal to 3/2 cosine of something x.
- And we could use the formula that we learned in the previous
- video, that it equals 2pi over the period x.
- So now we just have to look at the graph and try to figure
- out what the period of the graph is.
- Well how many radians does it take for the graph to
- start repeating again?
- Let me click on the hint button, and maybe
- this'll help us.
- When I click hint-- there; drew the period.
- And you could have figured it out on your own.
- If you just go from any point and then follow the curve back
- to the same point again, you'll see how long it's period is.
- And the hint on the Khan icon actually told us
- that the period is 4pi.
- And you could just start from any point to any other point.
- You could have gone from this point and then gone down,
- gone back up, come down.
- And then you would have seen that this distance is also 4pi.
- So we know that the period is 4pi.
- And then I could click hint again and it'll tell us stuff
- that we already figured out.
- The amplitude, we already figured out, was 1 1/2.
- But let's just use a period, because we already knew
- what the amplitude is.
- So the period here we already figured out was 4pi.
- So let's just write that in our equation.
- So f of x is equal to 3/2 cosine of 2pi divided by
- the period-- the period in this case is 4pi-- x.
- That equals 3/2 cosine of 1/2 x.
- Now if you ever forget this formula, which frankly
- I always do forget it.
- I've actually never memorized it.
- I just try to think about what the period would be.
- The way I think about it is the coefficient on the x-term,
- that's a measure of how many cycles does the graph
- do within 2pi radians.
- Let me see if I can explain that within the context
- of this problem.
- So this problem, if we start at 0 and then 2pi is here, how
- many cycles do we complete by the time we get to 2pi?
- We start here, we go back here, and then we're at 2pi.
- Well we only got halfway done through a cycle.
- So that's the coefficient on the x-term.
- And that's how I remember it.
- So I could just say, well that's 3/2 cosine of
- how many cycles do I complete in 2pi radians?
- Well I only complete half a cycle.
- 3/2 times the cosine of 1/2 x.
- So that's our f of x.
- Let's do another problem.
- All right.
- OK.
- This one's interesting.
- So the first thing, just by inspection we can figure out
- what this amplitude is.
- This is pretty easy.
- Right?
- How much does it move above and below the x-axis?
- Well it only goes 1, so we know that the coefficient, or the
- multiplier times the sine or the cosine function--
- whichever this is-- is 1.
- So let's write that down.
- Let's write down that the amplitude is equal to 1.
- Now let's try to figure out if this is a sine
- or a cosine function.
- In the last problem we said sine of 0 is 0
- and cosine of 0 is 1.
- Well f of 0 of this function, whichever it is, is 0.
- So we know this is a non-shifted sine function.
- So there.
- We have another piece of information.
- It's a sine function.
- So the last thing we have to figure out, we can either
- figure out the period or we could use the method that I
- just showed you where we say, well how many times does it
- cycle within 2pi radians?
- So let's do it that way.
- And then we immediately know the coefficient.
- Let's see.
- Well actually, this graph board doesn't even get
- all the way to 2pi.
- But let's see.
- It goes one cycle, two cycles.
- And I did two cycles in only pi radians, right?
- Because I'm only at x equals pi here.
- So if I did two cycles in pi radians, then we must be able
- to do four cycles in 2pi radians.
- Or we could actually start here.
- Actually, this is better.
- Right?
- Going from negative pi to pi.
- That's 2pi radians.
- So we finish one cycle, two cycles, three
- cycles, four cycles.
- So then we know what the coefficient on the x-term is.
- So we know that it is sine of 4x.
- So the answer here is f of x is equal to 1-- because that's the
- amplitude-- the amplitude times sine of 4x.
- I think we have time for one more.
- And I want you to-- don't just mechanically do whatever
- I'm telling you.
- I want you to think about why counting the number of cycles
- within 2pi radians, why that makes sense to you.
- Then think back to the unit circle.
- Or think back why that formula, the 2pi divided by the period,
- is also the coefficient.
- Think about why that makes sense.
- If you realize why it makes sense you'll never have to
- memorize it, and then 20 years later when you're doing it like
- I'm doing it right now you won't be confused.
- You'll be able to re-derive the formulas.
- Let's do one more.
- All right.
- So what's the amplitude here?
- Well, let's see.
- The amplitude is 1/2.
- So let me delete the old stuff that I was writing before.
- For some reason it's not deleting.
- OK.
- Hope I don't confuse you.
- So the amplitude, let's just call it Amplitude,
- is equal to 1/2.
- And how many cycles does it complete within 2pi radians?
- Let's see.
- If we start here it looks like it completes only half a cycle.
- Right?
- Because it takes 4pi radians to complete the entire cycle.
- So it only completes half a cycle.
- So we could think of it either two ways.
- We could say that the period is equal to 4pi, because that's
- how long it takes to complete one cycle, or we could say
- it can only complete half a cycle within 2pi radians.
- The last thing we have to figure out: Is it a sine
- or a cosine function?
- Well, f of 0 is 0.
- Right?
- So it's a non-shifted sine function.
- So then we're done.
- We have f of x is equal to 1/2-- figured out it's a sine
- function-- sine of what?
- How many cycles did it complete in 2pi radians?
- It only completed half a cycle.
- Let me not cover the problem.
- It only completes half a cycle.
- So it's 1/2 sine of 1/2 x.
- Or we could use the formula f of x equals goes 1/2 sine of
- 2pi divided by the period x.
- Right?
- Because 2pi divided by the period is equal to 2pi over
- 4pi, which also equals 1/2.
- I think that'll give you a sense of how to do
- these problems now.
- And I encourage you to practice them on the
- Khan Academy website.
- Have fun.
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