Break out the calculators, it's time to use the little machine to do something other than the easy basic math problems it's probably used to!
We will start here by going into the scenarios of when you would want to use each function, and what the value you get does for you. Before we drive right into the usage of the three main trig functions, we should make sure that everyone knows what we'll need to actually use those functiona effectively. Let's get started...
The above shows how to locate the opposite, adjacent, and hypotenuse of a triangle. Note the inner variable, X. The opposite and adjacent sides depend on the angle you are trying to find! If the angle you were looking for (denoted by the arc inside the triangle, and the Theta sign (circle with line through it) was the TOP angle of the triangle, the adjacent and opposite angles would be reversed.
As you can see, the opposite and adjacent switched because of the angle you are solving for. Think of it like this: The adjacent side is the side that makes forms the angle you are solving for; the opposite side is the the side that is not part of forming that angle!
Note: Graphics are not to scale!
When you find the sine of an angle, what you are also finding is the ratio of the opposite and hypotenuse of the triangle. As such:
sin(any angle) = opposite/hypotenuse
This means that, if you know one side and an angle, you can find out much more. Let's take a look at an example:
We want to find the value of X, the opposite side. First, we get the sin of the angle: sin(30) = .5.
Since .5 is equal to 1/2, we will use that because it is easier to visualize.
Now, since sin = opposite/hypotenuse, we can fill in some values:
1/2 = X/7
And this is just like any ordinary algebra equation! Multiply both sides by 7 to get X by itself:
7/2 = X, or X = 3.5.
Since the sin formula involves three values, if you know two of them, you can figure out the third! Now, let's take a look at Cosine...
Using COS is much like using SIN; if you have two know values, you can find the third one. However, COS can find different values than SIN can. Here is what COS is all about:
cos(angle) = adjacent/hypotenuse
Again, this has the same implications as the sin functions, as shown below.
Here, we first computer the COS of 45, which is .707. So, our equation turns out to be: .707 = X/9.
Multiply both sides by 9, and we have: 6.363 = X. And that is the length of the adjancent side!
And as you may have guessed, using the tangent function is very similar...
Of course, the tangent function provides the same functionality to find missing values in a triangle, but is used for different sides.
tan(angle) = opposite/adjacent
And once again, we see that the function is simply a relationship between the length of the sides and the degree of the angles.
Here, our equation is: tan(45) = 4/X
This one is easy, since the tangent of 45 is 1. So our equation ends up as: 1 = 4/X.
We get X by itself by dividing by 4, and get X=1/4. So the missing side we have is 1/4. (Be sure to remember the units. 1/4 may not seem like a lot, but if the units are in miles, we have the adjancent angle equal to 1/4 mile!)
And that's all there is to SIN, COS, and TAN. You are able to do many things with these functions, and the above explanations should get you started off on the right foot! Remember: these functions are simply ratios that you can use to find missing values in a triangle!
Ready for more? Head over to the Math Lesson Center and choose another topic!