Trigonometry, do you know when you will use trigonometry?
NEVER. (laughter)
You will never use it. It was invented to destroy the hopes and dreams of young people. (laughter)
So, its actually useful in things like physics. So, when you take your physics class or when you become a doctor you will need trigonometry.
I don't think it shows up in chemistry, but definitely for those of you who plan on becoming engineers you will need to know trigonometry.
So what is this trig stuff? Most of you have seen this already. You may remember the unit circle.
So, lets do it again. So first we do what is called right triangle trigonometry.
The idea is the following:
Thousands of years ago and you are ancient Babylonians and you are making war on the Hittites or the Philistines or whoever, and you want to knock down their wall...
Because they have a big wall around their city, and you could knock it down and get in there and get all of their loot. Right?
So, the goal, however, is to not get shot by the flaming arrows.
You want to come up with a way to knock down the wall. The trick is...
How can you figure out how high the wall is and how far away you are?
When trigonometry was invented people realized that it was sort of a ratio and it stayed the same no matter where you were.
You remember in geometry, there was a rule of similar triangles.
The rule was that the sides are proportional.
Say you have that little picture right there. Let's just call this angle, 20 degrees.
It doesn't matter what it equals.
So you are standing here.
The wall is over there and you want to figure out how high that wall is.
So people measured this distance.
People measured that distance a long time ago without getting the flaming arrows.
They stood back and knew that if you shot an arrow, then you would know how far that arrow was going. because they were good at that.
So you figure out how far away you are from something.
And, you figure out how tall this was.
Then they realized because it is similar triangles.
That this ratio was the same as this ratio.
In other words, y/x is the same as B/A
In fact, you can even create a little triangle. The ratio is the same as C/D.
And so on, they discovered that all of these ratios are the same.
So, once you had an angle. This ratio never changed as long as you were dealing with similar triangles.
Okay?
By the way, how do you know that triangles are similar? We learned this in geometry.
Triangles are similar, if among other things, if all the angles are equal.
This triangle here has a 20 degree angle, and a 90 degree angle.
That means the other one has to be 70 degrees.
And, this one has a 20 degree and 90 degree angles.
So, that has to be 70.
So, those triangles have to be similar.
And, once they are similar, you know that the sides always have the same ratios.
So, they gave names to these ratios.
The ratios they named, they named them sine, cosine, and tangent.
So, that is what we are going to learn.
The first most important ratio to learn is called sine.
So, if you have any triangle and you have an angle, which we will call x,
Then the sine of x, written: sin(x), is the ratio of this side (A) to that side (C).
The cosine of x, written: cos(x), is the ratio of B/C.
There is an angle mnemonic to remember these things.
Tangent of x, written: tan(x), is A/B.
You see the muscles there, that is called the gun show. (giggles)
Those are the three ratios that you will need.
Whats's our mnemonic?
I bet most of you have learned SOH CAH TOA.
They taught you in school. It goes like this.
What does it stand for?
This says that sine of any angle is the length of the side opposite divided by the length of the hypotenuse.
This needs to be sine of an angle because you, technically, cannot just have sine. It needs to be sine of x.
The cosine of the angle is the length of the adjacent side divided by the length of the hypotenuse.
The tangent of the angle is the length of the opposite side divided by the length of the adjacent side.
What do we mean by opposite and what do we mean by adjacent?
Say you are standing there and you are measuring up the wall. That wall is opposite you.
When I look at the wall, it is opposite me.
When you think of this angle, this is the opposite side.
The hypotenuse is the side opposite the 90 degree angle.
This is the hypotenuse.
The adjacent is the other side.
It is adjacent to me. I can touch it.
But, basically it is the leftover.
The third side is the adjacent.
The sine of any angle is the opposite divided by the hypotenuse.
The cosine is the adjacent side over the hypotenuse.
The tangent is the opposite side over the adjacent side.
For example,
Lets say I have some random triangle and this is the angle x.
This side is length 6.
This side is length 8.
First of all, if I know those two sides, I can learn how to find the hypotenuse. Right?
There is two formulas that you memorized in class.
You memorized the quadratic formula. Remember that one?
x= -b + or - the square root of b squared minus 4ac all over 2a.
There is a little song.
Then, there is the Pythagorean Theorem.
You guys don't know any others, right?
Maybe you do, but probably not.
But, I bet you all know the Pythagorean Theorem.
Right? Which is a^2 +b^2=c^2, of course, depending on where a, b and c are.
Ok, so...
We only need to be given 2 sides of a triangle.
You can, th en, always figure out the third.
So, if this is 6 and this is 8. Then, you can find that side.
That's very impressive.
6 squared plus 8 squared is c squared.
So, 36 plus 64...
100 is c squared.
So, c is 10.
c could also technically be negative 10,
but, in geometry we don't use negative numbers.
They don't make much sense.
So, we stick with positive 10.
Ok? But, now that we know that this is 10,
We can find the sine and cosine and tangent of x.
Sine of x is the opposite side, 6,
Remember, you look at the angle and look at the side opposite it and get 6.
Divided by the hypotenuse.
Cosine of x is
the adjacent over the hypotenuse of an angle.
Sure.
Okay. Sure.
Remember that.
8 over 10
Its the other side over 10.
And, then, the tangent
is the opposite over the adjacent.
You do not need to simplify these.
If this were a test and you wrote 6 over 8 that's fine.
You do not need to reduce it to 3 over 4.
Okay?
You can if you want. You have no obligation to reduce.
Webassign will reduce for you automatically.
Okay, so if you have something horrible,
on Webassign, 196 over 300,
You don't need to reduce that. Okay?
This reduces to
49 over 75.
Okay? But, you don't actually need to reduce it. You can just put it in that form.
Be careful with decimals in Webassign.
Because, you know, you might not be exactly right.
People have a tendency to round their decimals too much.
Oh! That reminds me.
You can use a calculator for any part of this class, but not in exams.
Um, anyway. So,
you do not need to reduce things.
These are numbers. We are going to try to keep the numbers nice and simple and not messy. Okay?
Um, in fact if you are doing a problem
and getting really bad numbers, you are probably doing it wrong.
Okay?
These should come out as very straight forward numbers.
If you are not sure,
just kind of circle it and say this is kind of what I am doing.
and you will receive partial credit.
And, at least you will get that part correct.
Okay, let's have you guys do another one of these.
Ah!
See that. That's the guns. (laughter)
Um, see you sit in the front and you get extra attention.
Its really good. You got to remember not to do that again.
Don't make that mistake twice.
Let's say that we give you x, and
this is, ah, 12
and that is 13.
You have to find the sine, cosine, and tangent.
Alright, so, to find the sine, cosine, and tangent.
First thing you do, you need to figure out the missing side.
So, you do the Pythagorean Theorem.
So, you say, A squared plus 12 squared is 13 squared.
Use your calculator.
Okay? So, if you solve this,
You get A equals
5
Okay? so this is a 5, 12, 13 triangle.
So far, so good?
Alright, now to find the sine...
You do opposite over hypotenuse
So, you do 12 divided by 13.
Did you get that one?
Yay! Because you did this in high school, right?
Okay. Cosine...
You do the adjacent over the 13.
Then, tangent. You do the opposite over the adjacent.
Simple?
What about using this angle, y?
Now, find the sine, cosine, and tangent.
The sine of y is
would be 5 over 13.
Notice
its the same thing as the cosine of x.
The sine of one angle is the cosine of the other angle.
That's what the co actually stands for.
Co means complementary.
Remember, from geometry, complementary angles are angles that add to 90 degrees.
So, these 2 angles have to add to 90 degrees.
The angles in triangles add up to 180.
And, that's 90.
So, one angle is the complement to the other.
So, the sine of one is the cosine of the other.
Okay? The cosine of y is 12 over 13.
Because if you look at y and say adjacent
over hypotenuse.
Tangent of y
is 5 over 12.
So far, so good?
Ah. Will we be ever asked to do sine, cosine, and tangent of the right angle?
No. Okay.
Because it doesn't really make sense.
Okay.
Remember when I told you that the sine of x is
opposite over the hypotenuse.
Sine of x is
A over C.
Cosine of x
is B over C.
There is a relationship there in the Pythagorean Theorem that is coming. Okay?
What if I put these on a circle instead?
I put them on
Something like that.
Okay? And I said,
Let's just pick some random coordinate.
I guess I should just call them x and y.
Alright, so that's
a coordinate x and y.
The coordinates of this point
are (x,y).
If I make that just 1,
What do I know about the sine of this angle down here?
Which we use the greek letter theta.
Do you know why we use theta?
I don't know. We just use greek letters.
I guess we did not want to use the other letters.
Okay?
We use the greek letter theta
to stand for the angle that is made by
the radius and the x-axis.
The sine of that angle
is y divided by 1.
Do you guys see that?
Let's blow it up.
Make it a little bigger.
The radius of this is 1.
And, if you pick any point on this circle,
this circle is called the unit circle.
Unit just means 1.
Okay?
The coordinates are x and y.
Because the coordinates of any point are x and y.
Think about the opposite for a minute.
You say the sine of theta
is the opposite side which is y
divided by the hypotenuse.
The hypotenuse is just 1.
So, the sine of theta is just going to be y.
The cosine of theta is just x.
Okay?
So in other words,
if I know the coordinates of a point,
on that circle, then you know
the sine and cosine of that angle.
This will be very handy
if I want to find sine and cosine for angles
that don't sit inside of a right triangle.
The problem with a right triangle is that it only goes up to 90 degrees.
So, how do you find sine and cosine
bigger than 90 degrees?
You do this.
What else do we notice about the unit circle?
I going to have to pull that back down.
If you have a point in the first quadrant,
what do you know about the coordinates of those points?
They are both positive.
Okay? So,
the x-coordinate and y-coordinate are both positive in the first quadrant.
In the second quadrant, over here.
The x-coordinate is now negative.
Because we are going to the left. The y-coordinate is still positive.
Okay, so
the coordinates would be
a negative value and a positive value.
Down here the coordinates
will be a negative value and a negative value.
And, down here
positive and negative.
So, remember the x-coordinate
is your cosine
and the y-coordinate is your sine.
So, this tells you
That in the first quadrant,
sine and cosine are both positive.
In the second quadrant,
the sine
will stay positive
but the cosine will be negative.
Why will it be negative?
Because you will be going this way.
And, x-coordinates will be negative over here.
And, the x-coordinate is cosine.
We will be doing a lot more of this on Wednesday.
So, don't be scared.
In the third quadrant,
they will both be negative.
In the fourth quadrant,
the sine is negative
and the cosine is positive.
Okay?
So, if I tell you what quadrant
a point is in
which we will figure out more of this on Wednesday.
Then, you can tell if it is positive or negative.
So, for the moment we are going to stick
with the first quadrant, where everything is positive.
I am going to teach you one other right triangle trigonometry for today.
That's what I am going to do for the rest of the day.
One of our favorite triangles is the 30, 60, 90 triangle.
Do you know why we use the 30, 60, 90 triangle?
Well,
If we have an equilateral triangle,
You know that's 60 degrees, that's 60 degrees, and that's 60 degrees.
Right? An equilateral triangle.
So, you cut it in half like this.
You get a,
you cut the triangle directly in half
perpendicular to the base
you get a right angle.
This angle becomes 30 degrees and that is 30 degrees.
And, you get a 30, 60, 90 triangle.
Okay, 30 degrees, 60 degrees, 90 degrees.
Furthermore,
This is an equilateral triangle,
so this distance
is half of this distance.
So, if this is x
The hypotenuse is 2x.
Okay?
In an equilateral triangle, that is 2x
and that is 2x.
And, if you use the Pythagorean Theorem.
This side
is x times the square root of 3.
That is a 3.
So, that is a 30, 60, 90 triangle.
And, those ratios never change.
So,
If this is a 30, 60, 90 triangle
and this is x that is going to be 2x.
and that is going to be x times the square root of 3.
That comes from the Pythagorean Theorem
and the fact that we are cutting an equilateral triangle in half.
So, let's find the sine, cosine, and tangent of all of this.
Okay. Sine of x
will be opposite divided by the hypotenuse.
So, x over 2x.
Which will be 1/2.
Cosine of x
is the x square root of 3 over 2x.
Which works out to the square root of 3 over 2.
Tangent of x
will be x over x times the square root of 3.
Which will reduce to 1 over the square root of 3.
I don't know why I wrote x. These should be sine, cosine, and tangent of 30 degrees.
That is why you don't do things in ink.
So, you use pencil.
Okay?
So, these are very handy to memorize.
Part of what we will do on Wednesday is to come up with a good way to memorize this.
Its easy. We will have to draw a triangle.
The sine of 30 degrees will always be 1/2.
The cosine of 30 degrees
will always be the square root of 3 over 2.
And, tangent of 30 degrees.
will always be 1 over the square root of 3.
Remember what I told you before about the sine and the cosine.
The sine of this angle
is the cosine of that angle.
So, the sine of 30 degrees
is the cosine of 60.
So, the sine of 60 degrees.
is the x root 3 over 2x.
So that is going to become square root of 3 over 2.
The cosine of 60 degrees
is 1/2. See how those are switching places?
And, the tangent of 60 degrees
Well, let's see. Tangent
is the opposite over the adjacent.
which is
x root 3 over x.
Which is just root 3.
Okay? So...
these are things you should memorize.
So far, so good?
Questions?
One last thing.
Okay. We can have
this 45, 45, 90 triangle.
That comes out of a square.
In a square you have a diagonal.
These 2 sides are the same.
And you do the Pythagorean Theorem,
That comes out x root 2.
Now, the sine of 45
will be x over x radical 2.
Also, known as
1 over radical 2.
And, if you want to show off,
you can rationalize that to be
radical 2 over 2.
Cosine of 45
will also be x over x radical 2.
So it will also be
1 over radical 2
or radical 2 over 2.
Tangent of 45
well, tangent is opposite over adjacent.
Notice the sides are the same.
it will be 1.
So, this class is not review for everybody.
Just for some.
What you want to do...
And, I really recommend you do this.
So, when the exam rolls around.
Because, remember, you will be nervous.
You will want to have a way to instantly
know the sines, cosines, and tangents.
You make a little grid.
30, 45, 60 across the top.
Sine, cosine, and tangent down the side.
Okay.
Sine and cosine all have 2 as their denominators.
Sine goes 1, 2, 3.
1, 2, 3
Great.
Cosine
goes 3, 2, 1
Just like that. Did anyone learn that in school?
Yeah, some of you. Okay.
Okay, good.
Not enough of you.
This is the way that it was taught to me in school.
Because you guys learn the unit circle
you kind of stand there and have to draw it in your head.
And you have no idea what you are doing.
But, if you can memorize this,
you can write this down for the exam.
Put it on the corner of your paper and you will be fine.
Tangent
is found by taking sine, and dividing it by the cosine.
So, the denominators will cancel.
So, this will come out 1 over the square root of 3.
square root of 2 over 2 over square root of 2 over 2 will come out 1.
Square root of 3 over 1
is the square root of 3.
Tangent, the tangent is found by taking the sine and dividing it by cosine.
You take the numerator of this and divided it by the numerator of this.
You get 1 over the square root of 3.
Okay?
And, on Wednesday, I will show you why tangent is sine over cosine.
You can kind of group all this in one.
Questions?
Guys, just feel free to ask questions.
No?
Well, we will get to more of these. So, the question is "what if it is not one of these angles".
So, for the moment, we will just deal with these,
because no calculator and we
are just starting out.
So, on that note it is enough for the first day.