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
More trig graphs Determining the equations of trig functions by inspecting their graphs.
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- So now we have this graph of this-- what was
- clearly a trig function.
- And our task is to figure out what the function is.
- So let's look at this.
- The first thing I do when I look at something like this,
- I want to figure out its period and its amplitude.
- So what's its amplitude?
- That's always an easy one.
- So the amplitude.
- Well, that's just how much does it move up and down above
- and below the x-axis?
- Well, the amplitude here is how much it moves up the x-axis.
- Well, it moves up 1/2 above and below the x-axis.
- So the amplitude is 1/2.
- And keep in mind, the amplitude is not this whole distance.
- It's not this.
- It's just how much it moves above or below the x-axis.
- So in this case, the amplitude is 1/2.
- And then we have to figure out what the period is.
- The period is, how long does it take-- how many radians does
- it go through for it to go through 1 complete cycle?
- Well, if we start here, and we were to follow the graph, it
- doesn't take until right here until we complete
- 1 cycle, right?
- Because here, we're still going down, and now
- we're going below.
- So we're not repeating yet.
- And here's where we start repeating again.
- And then we start repeating again here.
- So every pi radians, we start the cycle over again.
- It happens the same if you go backwards into the
- negative radians.
- So the period is pi, right?
- The period is pi.
- And you could start from any point.
- You could start from this point.
- And if you go, follow the graph, and then come back to
- the same point again, we see once again that the
- period is pi radians.
- Now we have to figure out if this is a sine or
- a cosine function.
- And for now, we'll not think about shifting.
- So let's think about what happens when-- you know,
- we want to know what this function is.
- f of x is equal to question mark.
- Well, we see that f of 0 is 0.
- f of 0 is equal to 0.
- What does that tell us?
- Is this a sine or a cosine function?
- Well, what's cosine of 0?
- Cosine of 0 is 1.
- And what's sine of 0?
- Well, sine of 0 is 0.
- And this function is 0.
- So we know that this is a sine function.
- So we know the formula is going to take the form f of x, it's
- going to equal the amplitude times sine of 2 pi
- over the period x.
- And if we just substitute these numbers we just figured out, we
- know that f of x is equal to 1/2 sine of 2 pi over pi x.
- The pi's cancel out and you get f of x is equal
- to 1/2 sine of 2x.
- Let's define another function.
- Let's define g of x is equal to 1/2 cosine of 2x.
- What would have this looked like?
- Or what would have-- yeah.
- The grammar's a little difficult.
- I picked the wrong color, because f of x is
- actually the pink one.
- This is the one we have now.
- So actually, let me circle that.
- This is f of x.
- f of x is this one right here.
- And now, g of x, I'm going to do--.
- So when x is 0, what is g of 0?
- Let's put 0 in here.
- So this whole term will become 0.
- What's cosine of 0?
- It's 1.
- And 1 times 1/2.
- So g of 0 is 1/2.
- So we would start here, and then we would have-- just
- like the sine function-- we would have a period of pi.
- Because it has the same coefficient here.
- So this'll just look like this.
- I think you get the point.
- It's just like the sine function was just shifted
- to the left of it.
- Well, I'm getting confused on this-- ignore this.
- But if you look at this side, the important thing to realize
- is that it intersects the y-axis at not 1, but 1/2.
- And the reason why it doesn't intersect it at 1, even
- though cosine of 0 is 1, is because we have this 1/2
- coefficient right here.
- I guess you can't call that a coefficient.
- I guess it's a 1/2 times the cosine function.
- Hopefully that gives you a little bit more of a sense of,
- if you just looked at a graph, being able to figure
- out its equation.
- And I'll actually do one more video where we'll actually use
- the Khan Academy trig graphing module to figure
- out a couple more.
- 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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