Graphs of trig functions Exploring the graphs of trig functions
Graphs of trig functions
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- Well what I want to do now is actually I'm going to use this
- graphing application to explore the trigonometric functions or
- explore the graphs of them.
- And just to start off or just to let you, this application
- I'm using is from gcalc.net.
- it's G C A L C .net.
- It's not mine, but I want to give them credit because
- that's what I'm using and I hope they don't mind.
- So let's start off just graphing some functions.
- Let's start off with the sine function.
- So let's say sine of x.
- I hope you all can see it.
- I'm typing it in up here.
- So sine x, let me graph that.
- Look at that.
- Like how nice that looks.
- So let's interpret this.
- So it's oscillating between-- well, let's
- just go point by point.
- I guess that's the easiest way to do it.
- So when x is equal to 0, what is the value of this function?
- Well, if we look here, the value of the function is--
- let me actually trace it.
- When x equals 0 and it has it written down at the
- bottom of this grey area.
- So when x is 0, y is also 0.
- And if you're remember when we looked for the definitions
- in the unit circle that's what we got.
- That the sine of 0 radians is 0.
- And now as we move on, or move along the curve, I
- have the trace function on.
- This is when x is equal-- if we look in the grey area at the
- bottom left it says 1.57.
- But what is that?
- If you're familiar with the-- 1.57 is more
- commonly known as what?
- It's 1/2 of what famous number?
- It's half of pi.
- We're at pi over 2.
- And if you want to convert pi over 2 to degrees
- that's 90 degrees.
- So when we're at the angle of pi over 2 radians the sine
- function is equal to 1.
- And if you go back to some previous modules you'll
- remember that that's exactly what the sine function was
- equal to when we looked at the unit circle.
- Because we were essentially at the point 1 comma 0.
- I hope it's not confusing that I keep referring to the unit
- circle that you can't see.
- But we'll keep going.
- But one thing I want to introduce here is the concept
- of the period or the frequency of the sine function.
- It's pretty obvious to you at this point that the function
- keeps repeating itself.
- It goes from 0, moves up to 1, goes back to 0, goes
- down to negative 1.
- Then goes back to 0 and then repeats again.
- So the period of this periodic function because that's what we
- call a function that keeps repeating, the period of this
- periodic function is this distance from here to here.
- And what's that?
- Well, that's 2 pi radians.
- And does that make sense?
- Well sure, because 2 pi radians is one complete revolution
- around the unit circle.
- And then it repeats again.
- And then it goes the other way.
- You go 2 pi radians backwards and things
- start repeating again.
- Pretty interesting, right?
- Oh, and another thing.
- What two numbers does it oscillate between?
- It oscillates between positive 1 and negative 1.
- And that makes sense because in the unit circle you can never
- get to a point on the perimeter of the unit circle that's
- larger than positive 1 or less than negative 1.
- And that's why the sine of x keeps oscillating between
- these two points.
- Let's do the cosine of x.
- Actually, I'm going to leave the sine of x there.
- It looks almost the same, but it looks shifted.
- It actually looks shifted to the left about
- pi over 2 radians.
- And that's actually the case.
- So let's first think about why.
- We figured out before that sine-- actually, it looks
- like this program is still tracing the sine function.
- That sine of 0 was 0.
- But if you look at the green function, the cosine of
- 0 radians is actually 1.
- Let me see if I can-- no.
- I don't know how to trace the cosine function, so
- I'll just do it here.
- The cosine of 0 is 1.
- And why does that make sense?
- Well, the cosine is the x-coordinate on
- the unit circle.
- When you have 0 radians or 0 degrees, you're at the point 1
- comma 0 on the unit circle.
- So 1 is the cosine or is the x-coordinate and
- 0 is the sine value.
- And if any of this is confusing, review the video
- where I use the unit circle to solve the various values of the
- trig functions and then this should make sense.
- And notice that this has a period similar to
- the sine function.
- It starts at 1, goes down to negative 1, and then
- comes back to positive 1.
- And it takes 2 pi radians to complete that cycle.
- And just like the sine function it's oscillating between 1 and
- negative 1 because on the unit circle you can't get to a point
- on the perimeter that's higher than that.
- And now to really hit the point home let's do
- the tangent function.
- I think this one might surprise you.
- Well, look at that.
- So the blue line is the tangent function.
- And why does it do this crazy thing?
- Well, if you remember, the tangent function is equal to
- the y over the x on the perimeter of the unit circle.
- Or since the y is the sine and cosine is the x, it also equals
- the sine over the cosine.
- So here, tangent is 0 whatever sine is 0
- because that makes sense.
- Because tangent is equal to sine over cosine.
- So it makes sense that when sine is 0, tangent is 0.
- But then, as the sine function becomes greater and the cosine
- function becomes less, the numerator in the tangent
- function becomes greater because the numerator is sine.
- So we get larger and larger values, all the way to the
- point where the denominator of the tangent function, which is
- the cosine function-- I think this is probably the most
- confusing module I've ever said because I can't really
- write these things down.
- The denominator goes to 0.
- The cosine right here.
- And then tan spikes and it actually approaches infinity.
- And if you look back at the unit circle, that actually
- might make a little bit of sense.
- But like the other functions, actually the tangent
- function has a period of pi instead of pi over 2.
- Instead of 2 pi.
- And I'll leave that as an exercise for you
- to think about.
- But with that drawn out, I'm now going to
- explore something else.
- Let me reset this.
- Yes, I really want to reset.
- I drew the sine function before.
- Let me draw the sine of let's say, 2x.
- Whoops, that's not right.
- sine of 2-- maybe I need to put some parentheses in.
- Oh, there we go.
- Actually, let me reset it.
- Yes, I want to reset.
- So first I'll draw the sine of x and then I'll
- draw the sine of 2x.
- So what's the first thing you notice about the difference
- between these two?
- The brown one is sine of x and the green one is sine of 2x.
- They both oscillate between the same two numbers and just so
- you know, the height of the oscillation is called
- the amplitude of this periodic function.
- So in both cases, the amplitude is 1 because they oscillate
- from 1 to negative 1.
- So the amplitude is 1, but their period is different.
- Sine of x takes 2 pi radians to complete one circle-- one cycle
- while sine of 2x only takes pi radians to complete one cycle.
- So it actually completes it twice as fast.
- And I want you to sit and think about why sine of 2x has 1/2
- the period of sine of x.
- And you can probably guess what happens if
- I type in sine of 3x.
- Actually, let's do sine of 4x.
- It should have 1/2 the period of sine of 2x then.
- And it does, even though this is probably a
- very confusing graph.
- So let's explore.
- So I think you understand what the coefficient
- on the x term does.
- When you have a larger coefficient it kind of
- speeds up the cycles.
- And let's explore a little bit more.
- Let's start off with sine of x again.
- And now, instead of making the coefficient larger, let's
- make the coefficient less.
- Let's make it sine of 0.5x.
- Look at that.
- Now, all of a sudden, it takes 4 pi radians to
- complete one cycle.
- And I want you to think about why that is.
- Because we're now slowing down how fast it cycles
- through the angles.
- Now I want to start playing with the amplitude.
- So we had sine of x, what do you think will happen if I put
- in this 2 times sine of x?
- So here, the period is the same.
- It's still 2 pi, but notice that it oscillates between 2
- and negative 2 instead of between 1 and negative 1.
- So whatever the coefficient, or whatever the number is in front
- of the sine or the cosine function, that actually
- affects its amplitude.
- And similarly, we can look at 0.5 sine-- let's
- say 0.5 sine of 2x.
- So now it only goes up to 0.5 and down to minus 0.5.
- So it's amplitude is 1/2 or 0.5.
- And it also oscillates twice as fast as the sine function
- because it was 0.5 sine of 2x.
- I think that's all the time I have now.
- I have a feeling this might have confused you more than
- helped, but I'll still put the video up just in case
- it's helpful for someone.
- But in the future I might actually record another video
- where I can actually write things down so it doesn't
- confuse you as much.
- So if it confused you I apologize, but I
- hope it was helpful.
- See you later.
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