THE BEGINNINGS OF TRIGONOMETRY:�THE SINE FUNCTION
OBJECTIVE
So, what is the Pythagorean Theorem?
This is the Pythagorean theorem:
Where each side of a triangle is: a, b, and c; and where a and b are the sides of the triangle with c as the hypotenuse, then:
Now, we can actually see this is true when looking at an actual triangle with measurements.
Like:
Now let’s plug in these numbers and see what we get:
a = 3
b = 4
c = 5
Great, but how do we know this is legit?
Just like I’ve told you guys, it isn’t enough for me to just show you what it is and expect you to take my word on it.
It’s important that you see why it works and how, so it makes better sense.
So, here’s the how and why it works.
1. First we need to start off with four equal copies of the same triangle.
(These are rotated 90, 180 and 270 degrees from the original as they need to be).
2. Next let’s set some sides (they will be labeled in a second).
Green – side c.
Red – side a
Blue – side b.
Now let’s maneuver them so it fits what we would like to show!
So now we have a square with a few properties
1. We have a square with a measurement of c on all sides
2. We have a smaller square in the middle with sides (a-b)
3. We know each triangle has an area of ½ ba
Now for some calculations.
We know the area of each triangle is ½ ba, so adding it up for all 4 we have: 4( ½ ab) which we know as 2ab
We know that the area of the little white square is (a-b) (a-b)
SO WHY IS THIS USEFUL?
This is useful because if we know two of the sides of a triangle, we can now find the third side without a problem.
We just need to know if the sides are just sides, or the hypotenuse.
This sounds more complicate than it is, so here is an example:
Example 1
Find the remaining side:
20
15
So, as we can see, we have two sides, and they want us to find the third.
First things first, we need to know if any of these sides is the hypotenuse of the triangle.�
If it is, then we need to make sure we put it as c.
In this case it isn’t, so we can just plug and chug!
So:
A = 15
B = 20
(No, it doesn’t matter what order, I just chose).
So, what we have is:
c = 25
Example 2:
Find the remaining side:
2
So, as we can see, we have two sides, and they want us to find the third.
First things first, we need to know if any of these sides is the hypotenuse of the triangle.�
If it is, then we need to make sure we put it as c.
In this case we can see that 2 is the hypotenuse, so we need to make sure we put that in its right place.
So:
So, what we have is:
So, that was a lot of review
So, that was quite a bit of review, but I wanted to make sure that you didn’t forget what we went over yesterday.
Now let’s go over the new material.
So to start, let’s begin with �SOHCAHTOA
SOHCAHTOA
A really weird acronym that will help you remember the basic trig functions
Let’s break them down:
S – Sine
O- Opposite
H- Hypotenuse
C- Cosine
A- Adjacent
H – Hypotenuse
T- Tangent
O- Opposite
A- Adjacent
So, what does this all mean?
It means this:
What does that mean?
Basically what we mean is when looking at a triangle, the angle that is being measured has certain sides associated to it.
But, this is easier to show than explain, so here is a triangle:
And here is the angle associated with that triangle
Now, the side that is opposite of this angle is:
The side that is adjacent of this angle is:
And of course, the hypotenuse of this triangle is:
So, in SOHCAHTOA, the sine of an angle is the opposite over the hypotenuse, or:
So, let’s see some examples:
Example 1:
20
35
13
Well, we remember from SOHCAHTOA that:
We can see that the side opposite the angle is 20
And, we can see that the hypotenuse is the biggest side, which we know is 35
So:
Or:
Example 2:
15
22
12
Well, we remember from SOHCAHTOA that:
We can see that the side opposite the angle is 15
And, we can see that the hypotenuse is the biggest side, which we know is 22
So:
Example 3:
16
27
10
Well, we remember from SOHCAHTOA that:
We can see that the side opposite the angle is 16
And, we can see that the hypotenuse is the biggest side, which we know is 27
So: