COSINE – FIND THE SIDE
OBJECTIVE
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:
NOW THAT THE REVIEW IS OVER
Let’s go over how to use cosine to find a missing side.
So, again, just like the sine function on your calculator is used to find the ratio that you know, but with certain angles, the cosine function does the same thing.
See, each angle has its own special proportion.
The adjacent side of that angle divided by the hypotenuse will always be a certain ratio, just like with Sine.
So, here are some examples to help:
Example:
15
30
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 30
So:
Or:
Finding the missing side using Cosine
So to find the missing side using cosine, we need to have two things.
We need an angle (and it’s actual measurement),
And we need either the hypotenuse of the triangle, or the adjacent side of the angle.
This is extremely similar to sine, just now we are looking for the adjacent side instead of the opposite side.
Example 1:
Find the missing side of the triangle if:
5.7
The hypotenuse = 5.7
So, we type into our calculators:
Then hit equal to get:
~0.707�Now we can set up the equation:
We know from SOHCAHTOA that:
Now we plug in what we know letting the unknown side be x:
5.7 * * 5.7
x = 4.03
Example 2:
Find the missing side of the triangle if:
3
The adjacent side = 3
So, we type into our calculators:
Then hit equal to get:
~0.866�Now we can set up the equation:
We know from SOHCAHTOA that:
Now we plug in what we know letting the unknown side be x:
x * * x
0.866x = 3
______ _____
0.866 0.866
x = 3.5
EXAMPLE 3:
Find the missing side of the triangle if:
9
The hypotenuse = 9
So, we type into our calculators:
Then hit equal to get:
0.5�Now we can set up the equation:
We know from SOHCAHTOA that:
Now we plug in what we know letting the unknown side be x:
9 * * 9
x = 4.5
Example 4:
Find the missing side of the triangle if:
2.7
The adjacent side = 2.7
So, we type into our calculators:
Then hit equal to get:
~0.259�Now we can set up the equation:
We know from SOHCAHTOA that:
Now we plug in what we know letting the unknown side be x:
x * * x
0.259x = 2.7
______ _____
0.259 0.259
x = 10.4
Example 5:
Find the missing side of the triangle if:
7.6
The hypotenuse = 7.6
So, we type into our calculators:
Then hit equal to get:
~0.087�Now we can set up the equation:
We know from SOHCAHTOA that:
Now we plug in what we know letting the unknown side be x:
7.6 * * 7.6
x = 0.66
One last look at the sides.
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 tangent of an angle is the opposite over the adjacent, or:
So, let’s see some examples:
Example 1:
15
20
35
Well, we remember from SOHCAHTOA that:
We can see that the side that is opposite of the angle is 15
And, we can see that the side that is adjacent to the angle is 20
So:
Or:
Example 2:
6
8
10
Well, we remember from SOHCAHTOA that:
We can see that the side that is opposite of the angle is 6
And, we can see that the side that is adjacent to the angle is 8
So:
Or:
BUT, WHEN I HIT SINE ON MY CALCULATOR….
SO, I’m sure that at least some of you have tried to figure out how to use the SIN button on your scientific calculator. �It’s not broken, you just didn’t know how to use it, but now I will show you.
The sine function on your calculator is used to find the ratio that you know, but with certain angles.
See, each angle has its own special proportion.
The opposite side of that angle divided by the hypotenuse will always be a certain ratio.
This is one of those times though that it is easier to show than to explain.
Example:
15
30
Well, we remember from SOHCAHTOA that:
We can see that the side that is opposite of the angle is 15
And, we can see that the hypotenuse is 30
So:
Or:
Finding the missing side using Sine
So to find the missing side using sine, we need to have two things.
We need an angle (and it’s actual measurement),
And we need either the hypotenuse of the triangle, or the opposite side of the angle.
Again, instead of explaining, here’s an example:
Example 1:
Find the missing side of the triangle if:
5.7
The hypotenuse = 5.7
So, we type into our calculators:
Then hit equal to get:
~0.707�Now we can set up the equation:
We know from SOHCAHTOA that:
Now we plug in what we know letting the unknown side be x:
5.7 * * 5.7
x = 4.03
Example 2:
Find the missing side of the triangle if:
3
The opposite side = 3
So, we type into our calculators:
Then hit equal to get:
~0.866�Now we can set up the equation:
We know from SOHCAHTOA that:
Now we plug in what we know letting the unknown side be x:
x * * x
0.866x = 3
______ _____
0.866 0.866
x = 3.5
EXAMPLE 3:
Find the missing side of the triangle if:
9
The hypotenuse = 9
So, we type into our calculators:
Then hit equal to get:
0.5�Now we can set up the equation:
We know from SOHCAHTOA that:
Now we plug in what we know letting the unknown side be x:
9 * * 9
x = 4.5