Where does the sign function come from we've all seen this function graphed before?

But where does it come from so I've plotted this graph in terms of degrees for demonstration purposes?

To understand whether where the sine function comes from we need to take a look at a circle?

I've plotted a circle here with a radius of 12 inches and

Inside of the circle I've inscribed triangles each of the triangles has a ninety degree Angle at its foot

And I've plotted them at different at different degrees

So this first one has an angle of 30 degrees relative to the x-Axis this one here

and then this one here has an angle of 45 degrees relative to the x-Axis and

Then this one here has an angle of 60 degrees relative to the x-Axis

so

Where the sine function comes from the sine function comes from plotting the height of each of these triangles?

Divided by the radius that is the hypotenuse of each triangle the hypotenuse of each triangle is the radius of the circle, right?

Because this is 12. This is 12

this is 12, so the hypotenuse is always the same for each of them and the sine function comes from

dividing the height of each triangle divided by that hypotenuse, so

Let's do the first one for 30 degrees. We will look at this triangle right here

Let's measure the height here, and we the height come is 6 inches

So 6 divided by 12 is 0.5. So we come over here. We look at our graph, and we come out to 30 degrees and

We see that 0.5. Is on the graph, Lo and Behold

So we contend that the sine of theta is simply equal to the ratio of the height of each respective

Triangle divided by the hypotenuse of each respective triangle, which is the radius so this is simply equal to

Y over R

so then let's do it for the next one so this one this one has an angle of 45 degrees relative to the x-Axis and

We measure the height of it and the height of it is eight point four eight inches

So eight point four eight divided by 12 is equal to 0.7 zero seven so we come over here 45 degrees

And we come up to the graph and Lo and Behold

It's at a it's at an elevation of 0.7 zero seven it has a y-value of point seven or seven

And then we come up, we would take a look at the if the triangle corresponding to sixty degrees

We measure the height of it and we get a height of ten point three nine

Ten point three nine inches ten point three nine inches divided by 12 is equal to 0.8 six six

So we come out to sixty degrees we come up to the graph and Lo and behold point eight six six is right there

And of course we see the number one because at ninety degrees

It's just the radius divided by the radius

because in ninety degrees the height of this triangle is is the height of the radius because we see that the

the width of this triangle is getting shorter and shorter and children until right here when the width of the triangle is equal to 0

And we progress around the circle plotting this ratio for all these different triangles

The one catch is that when we get to 180 degrees

We go negative

Because all of these triangles were above the x-Axis so these all had

positive y values positive heights that is but all of these have negative heights because they're below the x-Axis, so

We see that from 0 to 180 we have positive values

And we just plot all the way around the circle

But then when we get to 180 we go negative, and we have negative values

That is the ratios or Nega so let's just take a look at one of those

Take a look at this triangle right here

this triangle has an angle of

210 degrees

So it's 180

plus 30 which is 210 and so we measure the height of that one and

The height is 6 but it's below the x-Axis so it's negative 6 negative 6 divided by 12 is negative 0.5

so, Lo and behold we come out to 210 degrees you come down to the graph and

The whole look it's right there negative 0.5. Negative 0.5

And we continue plotting around the circle to get the rest of these values

That's it that is where the sine function comes from in the next tutorial we are going to discuss

What the utility of the sine function is and you know how can we use this thing?

How can we exploit this thing to to our advantage okay? See you next time