Basic Trigonometry - Khan Academy<br>An introduction to trigonometric functions: sine, cosine, and tangent.

00:05:13.000,00:05:17.500 Well once again, lets figure out what the adjacent side is. Well the adjacent side.

Sal: Welcome to the presentation on basic trigonometry!

Sorry it's taken so long to get a video out, but I have family in town

So lets get started with trigonometry.

Let me get the pen tool all set up. I'm a little rusty, I'll use green.

Trigonometry. I think it means...

tri, tri-go-nom-etry

I think it's from ancient Greek and it means triangle measure.

I think that's, I read it on Wikipedia a couple days ago so I believe that's the case.

But all trigonometry is, is really the study of right triangles

and the relationship between the sides and the angles of a right triangle.

Now that might sound a little confusing but I'll get started.

And actually you've probably seen a lot of these things we're going to go over now

and you'll finally know what those buttons on the calculator actually do.

So lets start with a right triangle.

Ah lets see, so its a triangle, and its a right triangle.

And lets do, just for simplicity. Lets say that this side is 3

this side is 4 and then the hypotenuse is 5.

So the trig functions tell you that for any angle, it tells you

what the ratios of the sides of the triangle are relative to that angle.

Let me give you an example, lets call this angle theta. Theta is the uh,

the Greek letter people tend to use for the angle you're going to find the trig function of.

Lets say that you wanted to find the sine,

and S I N is short for S I N E. (laughs)

Lets say you wanted to find the sin of theta.

So before we solve for the sine of theta, I'm just going to throw out a mnemonic that I remembered when I was learning this in school

and I carried, every time I do a trig problem I actually write down on the page

or at least I repeat it to myself.

And this is soh cah toa

some Indian princess who's name's Sohcahtoa, but I forget.

I have vague memories of my math teacher in high school telling a story about

But all you have to remember is soh cah toa.

Now you might say 'Well what is soh cah toa?'

Well soh cah toa says that sine is opposite over hypotenuse.

Cosine is adjacent over hypotenuse, and tangent is opposite over adjacent.

And that's going to be confusing right now, but we're going to do a lot of examples and I think it's going to make sense.

So lets go back to this problem, we wanted to know the sine of theta.

Theta is this angle in the triangle.

So lets go to our mnemonic, soh cah toa. So which one is sine?

Well S for sine, so we use soh. And we know that sine from this mnemonic.

Sine of say theta, is equal to opposite over hypotenuse,

opposite over hypotenuse.

Okay! So lets just figure out what the opposite and the hypotenuse are.

Well, what is the opposite side of this angle?

(sneeze) Excuse me. Well if we just go opposite the angle lets go here

The opposite side is 4, is this length of 4. I'll make that in a color.

So this side is the opposite. I'll circle it.

Now which side is the hypotenuse? You know this one,

you've been doing this in the Pythagorean theorem modules.

The long side, or the side opposite the right angle, is the hypotenuse.

So that is the hypotenuse. So now I think we're ready to figure out what the sine of theta is

The sine, whoops, I stayed in pink.

The sine of theta is equal to the opposite side, 4. Over the hypotenuse, which is 5.

We're done! Lets figure out, let me erase part of this

and we'll figure out some more things about this triangle. Let me erase this.

I think if you practice this you'll realise that this is probably one of the easier things you'll learn in mathematics

and its actually shocking that they wait till precalculus to teach this.

Because a smart middle schooler could, I think, easily handle this.

Not to make you feel bad if you're not getting it,

but just to give you confidence that you'll get it, and you will realize that it is very simple.

Lets go back to, okay! So lets figure out what the cosine

and C O S is short for cosine, I'll write it out cosine but you can. I'm sure you've seen it before.

So what is the cosine of theta?

Well we go back to our mnemonic, soh cah toa.

Well cosine is the cah. Right?

And that tells us that cosine of theta is equal to adjacent over hypotenuse.

Adjacent over hypotenuse.

The 4, this side, was the opposite side right, cause it's opposite the angle.

This side's the hypotenuse cause it's the longest side.

And just by deductive reasoning, but also just by looking at it you see that this side right here

the side of length 3 is adjacent adjacent to the angle, right.

Adjacent means, right beside it. So that's the adjacent side.

And we already figured out that the hypotenuse is that side that I wrote in pink.

So we're ready to figure out what cosine of theta equals.

Cosine of theta is equal to the adjacent side, that's 3, over the hypotenuse which is this pink side, 5.

Pretty straight forward isn't it? Lets do another one.

Okay, I don't want to erase the whole thing I just want to erase part of the page.

Cause I want to keep using this triangle. Lets see, lets seeee.

Lets see.

Okay, one left! The toa.

So if you remember what I said a little while ago, uh. Well we'll figure it out, but uh.

What is the ta, oh look how big that is.

What is the tan of theta, or the tangent of theta?

Well, lets go back to our mnemonic, toa right.

Toa is for tangent or t for tangent.

So it tells that tangent is the opposite over the adjacent.

So tan of theta is equal to opposite over adjacent. Well that equals,

well what was the opposite side? Right the opposite side was 4.

Right, and what was the adjacent side? Well we just saw that it was 3.

So the tangent of this angle is 4 over 3.

Now lets do another angle on this, lets call this angle here.

Lets call this angle here, um I don't know, lets call it x.

I don't know any other Greek letters. (laughs) Lets call that angle x.

So if we wanted to know the tan of x, lets see.

Lets see if its the same as the tan of theta.

The tan of x. Well now what's the opposite side?

Well now the opposite side is the white side, right?

Cause opposite this angle is the 3 side.

So we see here tan is opposite over adjacent. So opposite is 3, and then adjacent is 4.

This is interesting. The tangent of this angle,x, is the inverse of the tangent of that angle, theta.

I don't want to confuse you too much but I want to show you that

when you take the trig functions it matters which angle of the right angle you're taking the functions of.

And you might be saying 'Well this is all good and well Sal but what use is this?'

Well, I'll later show you that if you have some of the information,

say you know an angle and you know a side, or you know a couple sides, you can figure out...

And if you have a either a slide ruler or a trig table or a good calculator,

you can figure out um, given the sides of a triangle, you can figure out the angles.

Or given an angle and a side you can figure out other sides. And we're actually going to do that in the next module.

So hopefully this gives you a little introduction, I'm almost out of time on the YouTube ten minute limit.

So I'm going to wait to do a couple more examples in the next video. See you in the next presentation! Bye.

00:05:13.000,00:05:17.500 Well once again, lets figure out what the adjacent side is. Well the adjacent side.

Sal: Welcome to the presentation on basic trigonometry!

Sorry it's taken so long to get a video out, but I have family in town

So lets get started with trigonometry.

Let me get the pen tool all set up. I'm a little rusty, I'll use green.

Trigonometry. I think it means...

tri, tri-go-nom-etry

I think it's from ancient Greek and it means triangle measure.

I think that's, I read it on Wikipedia a couple days ago so I believe that's the case.

But all trigonometry is, is really the study of right triangles

and the relationship between the sides and the angles of a right triangle.

Now that might sound a little confusing but I'll get started.

And actually you've probably seen a lot of these things we're going to go over now

and you'll finally know what those buttons on the calculator actually do.

So lets start with a right triangle.

Ah lets see, so its a triangle, and its a right triangle.

And lets do, just for simplicity. Lets say that this side is 3

this side is 4 and then the hypotenuse is 5.

So the trig functions tell you that for any angle, it tells you

what the ratios of the sides of the triangle are relative to that angle.

Let me give you an example, lets call this angle theta. Theta is the uh,

the Greek letter people tend to use for the angle you're going to find the trig function of.

Lets say that you wanted to find the sine,

and S I N is short for S I N E. (laughs)

Lets say you wanted to find the sin of theta.

So before we solve for the sine of theta, I'm just going to throw out a mnemonic that I remembered when I was learning this in school

and I carried, every time I do a trig problem I actually write down on the page

or at least I repeat it to myself.

And this is soh cah toa

some Indian princess who's name's Sohcahtoa, but I forget.

I have vague memories of my math teacher in high school telling a story about

But all you have to remember is soh cah toa.

Now you might say 'Well what is soh cah toa?'

Well soh cah toa says that sine is opposite over hypotenuse.

Cosine is adjacent over hypotenuse, and tangent is opposite over adjacent.

And that's going to be confusing right now, but we're going to do a lot of examples and I think it's going to make sense.

So lets go back to this problem, we wanted to know the sine of theta.

Theta is this angle in the triangle.

So lets go to our mnemonic, soh cah toa. So which one is sine?

Well S for sine, so we use soh. And we know that sine from this mnemonic.

Sine of say theta, is equal to opposite over hypotenuse,

opposite over hypotenuse.

Okay! So lets just figure out what the opposite and the hypotenuse are.

Well, what is the opposite side of this angle?

(sneeze) Excuse me. Well if we just go opposite the angle lets go here

The opposite side is 4, is this length of 4. I'll make that in a color.

So this side is the opposite. I'll circle it.

Now which side is the hypotenuse? You know this one,

you've been doing this in the Pythagorean theorem modules.

The long side, or the side opposite the right angle, is the hypotenuse.

So that is the hypotenuse. So now I think we're ready to figure out what the sine of theta is

The sine, whoops, I stayed in pink.

The sine of theta is equal to the opposite side, 4. Over the hypotenuse, which is 5.

We're done! Lets figure out, let me erase part of this

and we'll figure out some more things about this triangle. Let me erase this.

I think if you practice this you'll realise that this is probably one of the easier things you'll learn in mathematics

and its actually shocking that they wait till precalculus to teach this.

Because a smart middle schooler could, I think, easily handle this.

Not to make you feel bad if you're not getting it,

but just to give you confidence that you'll get it, and you will realize that it is very simple.

Lets go back to, okay! So lets figure out what the cosine

and C O S is short for cosine, I'll write it out cosine but you can. I'm sure you've seen it before.

So what is the cosine of theta?

Well we go back to our mnemonic, soh cah toa.

Well cosine is the cah. Right?

And that tells us that cosine of theta is equal to adjacent over hypotenuse.

Adjacent over hypotenuse.

The 4, this side, was the opposite side right, cause it's opposite the angle.

This side's the hypotenuse cause it's the longest side.

And just by deductive reasoning, but also just by looking at it you see that this side right here

the side of length 3 is adjacent adjacent to the angle, right.

Adjacent means, right beside it. So that's the adjacent side.

And we already figured out that the hypotenuse is that side that I wrote in pink.

So we're ready to figure out what cosine of theta equals.

Cosine of theta is equal to the adjacent side, that's 3, over the hypotenuse which is this pink side, 5.

Pretty straight forward isn't it? Lets do another one.

Okay, I don't want to erase the whole thing I just want to erase part of the page.

Cause I want to keep using this triangle. Lets see, lets seeee.

Lets see.

Okay, one left! The toa.

So if you remember what I said a little while ago, uh. Well we'll figure it out, but uh.

What is the ta, oh look how big that is.

What is the tan of theta, or the tangent of theta?

Well, lets go back to our mnemonic, toa right.

Toa is for tangent or t for tangent.

So it tells that tangent is the opposite over the adjacent.

So tan of theta is equal to opposite over adjacent. Well that equals,

well what was the opposite side? Right the opposite side was 4.

Right, and what was the adjacent side? Well we just saw that it was 3.

So the tangent of this angle is 4 over 3.

Now lets do another angle on this, lets call this angle here.

Lets call this angle here, um I don't know, lets call it x.

I don't know any other Greek letters. (laughs) Lets call that angle x.

So if we wanted to know the tan of x, lets see.

Lets see if its the same as the tan of theta.

The tan of x. Well now what's the opposite side?

Well now the opposite side is the white side, right?

Cause opposite this angle is the 3 side.

So we see here tan is opposite over adjacent. So opposite is 3, and then adjacent is 4.

This is interesting. The tangent of this angle,x, is the inverse of the tangent of that angle, theta.

I don't want to confuse you too much but I want to show you that

when you take the trig functions it matters which angle of the right angle you're taking the functions of.

And you might be saying 'Well this is all good and well Sal but what use is this?'

Well, I'll later show you that if you have some of the information,

say you know an angle and you know a side, or you know a couple sides, you can figure out...

And if you have a either a slide ruler or a trig table or a good calculator,

you can figure out um, given the sides of a triangle, you can figure out the angles.

Or given an angle and a side you can figure out other sides. And we're actually going to do that in the next module.

So hopefully this gives you a little introduction, I'm almost out of time on the YouTube ten minute limit.

So I'm going to wait to do a couple more examples in the next video. See you in the next presentation! Bye.

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