0:00

[NOISE] >> Welcome,

Â in this unit weâ€™ll be learning about Trigonometry.

Â Trigonometry is the study of angles of triangles.

Â Letâ€™s look at the technical definition.

Â Trigonometry is a branch of mathematics that deals with relationships between

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

Â particularly involving trigonometric functions.

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

Â We want to study both triangles, circles, and

Â the relationships between the angles of these objects.

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

Â and we want to study oscillatory behavior.

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

Â Let's look at a simple example.

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

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

Â Notice if I take my Slinky and bounce it.

Â I get some oscillations.

Â The Slinky oscillates back and forth.

Â You can also take the Slinky and make some waves.

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

Â Notice that the polynomials we've been studying so

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

Â Polynomials can have some wiggles to them, but

Â eventually they either tend to infinity or minus infinity.

Â So we need some sort of function with periodic behavior.

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

Â over again, forever and ever.

Â 1:42

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

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

Â If we consider a circle of radius, r, and

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

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

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

Â Then we have the triangle labeled as we see here.

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

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

Â We get a unit circle and

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

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

Â There's nothing special about the letter theta.

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

Â 2:36

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

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

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

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

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

Â and the opposite side.

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

Â 3:03

This brings us to those trigonometric functions we mentioned before.

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

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

Â These three functions can be related to the right triangle

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

Â The cosine is the adjacent side over the hypotenuse and

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

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

Â Socahtoa basically is soh, sine is opposite over hypotenuse.

Â Cah, cosine is adjacent over hypotenuse and

Â toa, tangent is opposite over adjacent.

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

Â 3:51

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

Â any circle, for that matter.

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

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

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

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

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

Â degrees.

Â 45 degrees, 60 degrees, 90 degrees, etc.

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

Â on the unit circle are easy to find.

Â 4:31

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

Â trigonometric functions.

Â We also work something called radians.

Â Most of you probably know a circle has 360 degrees,

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

Â The relationship between degrees and

Â radians is simply that 360 degrees = 2 pi radians.

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

Â radian value of some key angles on the unit circle.

Â We can also look at the trigonometric values

Â of the trig functions at those key angles.

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

Â tangent with those angles, theta as the input.

Â Notice here we have some values.

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

Â It takes a little bit of work but

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

Â Being a master of trig values is super important.

Â 5:35

Notice on this table I snuck in three extra functions.

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

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

Â Sine, cosine, and tangent.

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

Â And another trig function,

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

Â 6:02

Let's look at our three main trig functions.

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

Â Just looking at a triangle,

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

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

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

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

Â the amount of time before that function repeats again.

Â 6:38

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

Â Trig identities are basically relationships between our trig functions

Â that help us solve equations and

Â manipulate expressions involving the trig functions.

Â There's quite a few of these.

Â Many of them you'll want to memorize.

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

Â figure out what the relationship between these quantities is.

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

Â 7:05

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

Â You may have seen this in a geometry course.

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

Â If you remember what that means for a triangle,

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

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

Â This is just an example to give you a flavor for

Â what these trig identities are all about.

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

Â might be helpful in evaluating some sort of equation.

Â In this unit,

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

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

Â forth between degrees and radians.

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

Â Do you want the smaller acute angle or

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

Â 8:26

Well applications of trigonometry include things like astronomy.

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

Â They wanted to figure out things like where

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

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

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

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

Â Other things include geography, optics,

Â architecture, mechanics, seasonal phenomena.

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

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

Â something that might be suitable for study with trig functions.

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

Â Because most signals you could think of as waves and

Â studying those waves involves trig functions.

Â Well, I hope you enjoyed learning about trigonometry.

Â Thank you and we'll see you next time.

Â