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[NOISE] >> Welcome,

in this unit we’ll be learning about Trigonometry.

Trigonometry is the study of angles of triangles.

Let’s look at the technical definition.

Trigonometry is a branch of mathematics that deals with relationships between

sides and angles of triangles, and the calculations based on them,

particularly involving trigonometric functions.

You can think of trigonometry as kind of having four components.

We want to study both triangles, circles, and

the relationships between the angles of these objects.

We look at three main trigonometric functions, sine, cosine and tangent,

and we want to study oscillatory behavior.

As we see in this graph here, the functions that oscillate back and forth.

Let's look at a simple example.

When you're a kid, you might have had a slinky toy.

Well, the slinky toy is a great example of oscillatory behavior.

Notice if I take my Slinky and bounce it.

I get some oscillations.

The Slinky oscillates back and forth.

You can also take the Slinky and make some waves.

When I make waves with my Slinky, that's another example of oscillatory behavior.

Notice that the polynomials we've been studying so

far, won't be able to account for this type of behavior.

Polynomials can have some wiggles to them, but

eventually they either tend to infinity or minus infinity.

So we need some sort of function with periodic behavior.

By periodic I mean, it oscillates with time and does the same thing over and

over again, forever and ever.

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Let's look at the correspondence between circles and triangles.

You might not think these two things have a lot in common.

If we consider a circle of radius, r, and

we think of a point on the circle with coordinates (x,y).

We could create a triangle from this circle by dropping a perpendicular

from the point (x,y) onto the x-axis.

Then we have the triangle labeled as we see here.

This is often called the unit circle when I replace that radius with one.

So take the circle I had before, replace the radius with one.

We get a unit circle and

we call the angle that this radial line makes with the x-axis an angle theta.

Theta is just a Greek letter that we often use for angle.

There's nothing special about the letter theta.

It's just a convenient way to let people know we're talking about an angle there.

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We can think about well, how do we determine what theta was

if we knew some of the sides from that right triangle.

So with my right triangle, I have three kind of main sides to the triangle.

There's the hypotenuse, the side opposite the right angle.

There's the adjacent side, the side next to the angle I'm interested in theta,

and the opposite side.

The side on the opposite side of the triangle from theta.

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This brings us to those trigonometric functions we mentioned before.

We have three main trigonometric functions we'll talk about.

That's the sine, the cosine, and the tangent.

These three functions can be related to the right triangle

via the relationship that the sine is the opposite side over the hypotenuse.

The cosine is the adjacent side over the hypotenuse and

the tangent of an angle is the opposite side over the adjacent.

A lot of students like to use the mnemonic soh cah toa to remember this.

Socahtoa basically is soh, sine is opposite over hypotenuse.

Cah, cosine is adjacent over hypotenuse and

toa, tangent is opposite over adjacent.

Let's take some more look at these trigonometric functions.

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Well we said that this triangle was also related to the unit circle or

any circle, for that matter.

The unit circle is just a special circle with radius one.

On that unit circle there are several angles we study a lot,

because these are values that we can evaluate explicitly for a circle.

So those points x y on the circle that are interesting or we know the values of.

Would be things going from the x axis along the unit circle to angles like 30

degrees.

45 degrees, 60 degrees, 90 degrees, etc.

These special angles are just values which the x,y coordinates

on the unit circle are easy to find.

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Well we don't always work with degrees in mathematics what we're looking at

trigonometric functions.

We also work something called radians.

Most of you probably know a circle has 360 degrees,

what you may not know is that we also can say a circle has 2 pi radians.

The relationship between degrees and

radians is simply that 360 degrees = 2 pi radians.

So in these two circles, I'm just depicting the degree value and

radian value of some key angles on the unit circle.

We can also look at the trigonometric values

of the trig functions at those key angles.

So for example, I can make a table of my sine, cosine, and

tangent with those angles, theta as the input.

Notice here we have some values.

By the end of this course, you definitely want to memorize this table.

It takes a little bit of work but

it will be worth it, particularly when you get to calculus.

Being a master of trig values is super important.

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Notice on this table I snuck in three extra functions.

I haven't talked much about these three extra trigonometric functions,

because they're basically directly related to the ones you already know.

Sine, cosine, and tangent.

The cosecant is 1 over sine, the secant of feta is 1 over cosine.

And another trig function,

the cotangent, is 1 over the tangent, or cosine over sine.

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Let's look at our three main trig functions.

I said that they correspond to oscillatory behavior, or an oscillating function.

Just looking at a triangle,

it may not be clear to you that those are oscillatory functions.

But basically as I change an angle, theta, and input it into this function,

I get differing values, and these values repeat with time.

Each of these functions has what we call a characteristic period, or

the amount of time before that function repeats again.

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Another thing that you'll be learning a lot about is trig identities.

Trig identities are basically relationships between our trig functions

that help us solve equations and

manipulate expressions involving the trig functions.

There's quite a few of these.

Many of them you'll want to memorize.

Some of them you'll want to make sure you know how to derive or

figure out what the relationship between these quantities is.

You'll be spending a lot of time working with trig identities.

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One trig identity you may have already seen is Pythagorean Theorem.

You may have seen this in a geometry course.

The Pythagorean Theorem says that a squared + b squared = c squared.

If you remember what that means for a triangle,

one side squared plus the other side squared gives you that hypotenuse squared.

The way we say this with trig functions is, sine squared + cosine squared = 1.

This is just an example to give you a flavor for

what these trig identities are all about.

There are relationships between your trig functions that give you an expression that

might be helpful in evaluating some sort of equation.

In this unit,

you're going to learn several things about trigonometric functions.

First of all, you're going to learn how to work with angles, convert back and

forth between degrees and radians.

And figure out how to figure out which angle are you talking about.

Do you want the smaller acute angle or

the larger obtuse angle when you're drawing these pictures?

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Well applications of trigonometry include things like astronomy.

Trigonometry was actually first developed to help talk about astronomy problems.

They wanted to figure out things like where

bodies were going to be in sky at different periods of time,

and that involves a lot of triangles if you think about it.

I know the planet was here one day and up there the next.

I'm making some sort of angle that I can use trigonometry to deal with.

Other things include geography, optics,

architecture, mechanics, seasonal phenomena.

Anything that changes with time in some sort of oscillatory behavior like

the seasons, or the temperature as a function of time throughout the year is

something that might be suitable for study with trig functions.

Signal processing is another very popular area to use trigonometry with.

Because most signals you could think of as waves and

studying those waves involves trig functions.

Well, I hope you enjoyed learning about trigonometry.

Thank you and we'll see you next time.