THE BEGINNINGS OF TRIGONOMETRY:�THE COSINE FUNCTION
OBJECTIVE
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:
Now let’s go back to �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 cosine of an angle is the adjacent 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 that is adjacent of 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 that is adjacent of 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 that is adjacent of the angle is 16
And, we can see that the hypotenuse is the biggest side, which we know is 27
So:
Now it gets tricky
Some of you may have noticed that the cosine of some angles are the same as the sine of others.
That’s because sine and cosine are very similar, and the cosine of one angle is the same as the sine of another. �So, when do we use them individually?�Well, that takes time to master, but basically with the right work, you can change them to suit your needs.
Here’s an example:
Example 1:
22
36
16
Well, we remember from SOHCAHTOA that:
We can see that the side that is adjacent of the angle is 22
And, we can see that the hypotenuse is the biggest side, which we know is 36
So:
Or:
Ω
We can see that the side that is opposite of the angle is 22
And, we can see that the hypotenuse is the biggest side, which we know is 36
So:
Or:
It’s all about perspective
You may find that you can find the answer by taking the sine of some angle.
You may also find that you can find the same answer by taking the cosine of some other angle.
Both are the correct answer.
To prove this, let’s try a few more:
Example 2:
12
19
16
Well, we remember from SOHCAHTOA that:
We can see that the side that is adjacent of the angle is 12
And, we can see that the hypotenuse is the biggest side, which we know is 19
So:
Ω
We can see that the side that is opposite of the angle is 12
And, we can see that the hypotenuse is the biggest side, which we know is 19
So:
Example 3:
27
35
21
Well, we remember from SOHCAHTOA that:
We can see that the side that is adjacent of the angle is 27
And, we can see that the hypotenuse is the biggest side, which we know is 35
So:
Ω
We can see that the side that is opposite of the angle is 27
And, we can see that the hypotenuse is the biggest side, which we know is 35
So:
So what are we getting at?
Well, what we know is the sine of one angle is the cosine of another angle.
But, let’s actually look at that more in depth.
Before we do, it’s important to note that we are working with right triangles.
This means that the triangle has a right angle in it.
Which means the rest of the angles in the triangle must add up to 90 degrees.
So, remember that as we go over this little example:
Example 1:
30
39
25
Well, we remember from SOHCAHTOA that:
We can see that the side that is adjacent of the angle is 30
And, we can see that the hypotenuse is the biggest side, which we know is 39
So:
We can see that the side that is opposite of the angle is 30
And, we can see that the hypotenuse is the biggest side, which we know is 39
So:
But wait, this actually seems a little different.
The sine of 50 degrees was equal to the cosine of 40 degrees.
So, that must mean that……….
The sine and cosine complementary relationship
In other words:
And
So, now knowing this relationship, let’s use this in some examples:
Example:
Well to find the complement of this sine angle, we just need to find its corresponding cosine angle.
So, to do so, we just need to subtract 90 from it and figure out what we would get.
So:
Or, in other words: