Graph of the sine function Using the unit circle definition of the sine function to make a graph of it.
Graph of the sine function
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- In the last presentation we kind of re-defined the sine,
- the cosine, and the tangent functions in a broader way
- where we said if we have a unit circle and our theta is, or our
- angle, is -- let me use the right tool -- let's say, and
- our angle is the angle between, say, the x-axis and a radius in
- the unit circle, and this is our radius.
- the coordinate of the point where this radius intersects
- the unit circle is x comma y.
- Our new definition of the trig functions was that sine of
- theta is equal to the y-coordinate, right, this is
- y-coordinate where it intersects the unit circle.
- And remember, this is the unit circle.
- It's not just any circle, which means it has a radius of 1.
- Cosine of theta is equal to the x-coordinate of this point.
- This is the x-coordinate.
- And tangent of theta equaled opposite over
- adjacent or y over x.
- That's interesting because that's also equal to sine of
- theta over cosine of theta.
- I'll just do that.
- I wasn't even planning on covering that, but just it
- leaves you something to think about.
- So with that said, let's take a look or let's try to see how
- this defines these functions.
- And I guess a good place to start is just with the sine
- function and we can try to graph it.
- So let's write, let's do a little table like we always do
- when we define a function.
- Let's put in values of theta, and let's figure
- out what sine of theta is.
- So when theta is equal to 0 radians, what is sine of theta?
- So when theta's 0, right, then the radius between it -- this
- is the radius and this is the point where the radius
- intersects the unit circle.
- And this point has a coordinate 1 comma 0, right?
- And so if where it intersects the unit circle is at 1 comma
- 0, then sine of theta is just the y-coordinate.
- So sine of theta is 0.
- If we said what is sine of theta when theta is
- equal to pi over 2.
- So now our radius is this radius and we intersect the
- unit circle right here at the point 0 comma 1.
- And what's the y-coordinate at 0 comma 1?
- Well it's 1.
- What happens when we have theta is equal to pi radians?
- So at pi radians we intersect the unit circle right here.
- We're at pi radian.
- This is the angle, pi.
- We intersect with unit circle at negative 1 comma 0.
- Because once again, this is the unit circle.
- So at negative 1 comma 0, what's the y-coordinate?
- Well, it's 0.
- So sine of pi is equal to 0.
- Let's just keep going around the circle.
- When we have the angle, when theta is equal to 3 pi over 4
- -- no, sorry, 3 pi over 2.
- Because this is pi and this is another half pi.
- So this is 3 pi over 2, sorry.
- So when theta is equal to 3 pi over 2, what is sine of theta?
- Well, now we intersect the unit circle down here at the
- point 0 comma negative 1.
- So now sine of theta is equal to negative 1.
- Then if we go all the way around the circle to 2
- pi radians, we're back at this point again.
- So sine of theta, so we're at 2 pi, sine of theta
- is now 0 once again.
- So let's graph these points out and then we'll try to figure
- out what the points in between look like, and I'll show
- you the graph of a sine function is.
- So let's draw the x-axis.
- This is my x-axis.
- And let's draw the y-axis.
- Not as clean as I wanted to draw it.
- This is y.
- And that's x.
- But in this case instead of saying that's the x-axis, let's
- call that the theta axis, because we defined theta as the
- input or our domains in terms of theta.
- So this is the theta axis.
- Now we're going to graph sine of theta.
- So when we said when theta equaled 0, sine
- of theta is equal to 0.
- So that's this point right here, 0 comma 0.
- When theta is equal to pi over 2, sine of theta is equal to 1.
- So this is the point pi over 2 comma 1, right?
- That's just this 1.
- When theta is equal to pi, sine of theta is 0 again.
- So this is the point pi comma 0.
- And when theta equaled 3 pi over 2, what was sine of theta?
- I equaled negative 1.
- Then when we got to 2 pi -- when we got to theta equal
- to 2 pi, sine of theta, again, equaled 0.
- So we know that these points are on the graph
- of sine of theta.
- And if you actually tried the points in between, and as
- an exercise it might be interesting for you to do so.
- You could actually figure out a lot of the points using
- 30-60-90 triangles or using the Pythagorean Theorem.
- But you actually get a curve that looks something -- let me
- use a nicer color than this kind of drab grey -- you
- get a graph that looks something like this.
- And you've probably seen that before.
- The term for this function is actually a sine wave.
- It looks like something that's oscillating or
- that's moving up and down.
- And actually if you were to put in thetas that were less than
- 0, the sine wave will keep going into the
- negative theta axis.
- It keeps going forever in both directions.
- It keeps oscillating between 1 and negative 1 and
- the points in between.
- So that's the graph of the sign function.
- In the next module I'll actually do the graph of the
- cosine function, or actually I might just show you the graph
- of the cosine function.
- Then I'll show you how they relate and how these can
- describe any kind of, or many types of oscillatory things in
- the world and how it relates to frequency and amplitude.
- So I'll see you in the next module.
- And just for fun you might want to sit down with a piece of
- paper and try to graph the cosine function or the
- tangent function as well.
- Have fun.
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