Amy has a master's degree in secondary education and has taught math at a public charter high school.
Law of Cosines
This video lesson will show you how you can use the law of cosines, a formula to help you solve all kinds of triangles, to help you find the missing side or angle of any kind of triangle. The formula looks very similar to the Pythagorean Theorem, a^2 + b^2 = c^2, with just one difference. Here it is:
The only difference between the law of cosines and the Pythagorean Theorem is that we have a minus 2ab cos (C). An easy way to remember this part is to remember a, b, and c again. We have a, b, and c again, so we have a 2 in front. See if you can think of other memory tricks to help you remember this formula.
Just like in the Pythagorean Theorem, our small letters a, b, and c stand for the sides of the triangle. But since this formula works for any kind of triangle, our letter c can be for any side of the triangle, not just the hypotenuse of a right triangle. The large letter C at the end stands for the angle C that is opposite side c.
In this video lesson, we are specifically looking at oblique triangles. These are triangles that are not right triangles. What kinds of triangles does this cover? It covers all kinds of triangles. It covers acute triangles, scalene triangles, obtuse triangles, and even equilateral triangles. Any kind of triangle that is not a right triangle is an oblique triangle.
Looking at the formula, we can see that it can help us to find the measurement of side c if we know the measurement of the other two sides, a and b, along with the angle opposite side c. We can also find the measure of angle C if we know the measurements of all three sides. We can solve for the measure of angle C by doing some algebraic rearranging of the formula.
Do you want to see a couple of examples of how this is done?
Let's look at this example, where we want to find the measurement of a missing side.
Find side c.
If this triangle wasn't labeled in any way, we could simply label the side we want as side c since that will make our formula easier to use. Because the formula works for any triangle, it doesn't matter which side we label with a, b, or c. We can label it any way that will make our problem solving easier.
In this case, the side we want to find is already labeled as side c, which helps us out a lot. We can go ahead and label the other two sides as a and b. Again, it doesn't matter which is which. So, I will go ahead and label the 7 as side a and the 10 as side b. Our angle C, the angle opposite the side we want to find, is 81. Now, we have a = 7, b = 10, and C = 81. We plug in these values into our formula.
Now we can evaluate the formula and then solve it. We get c^2 = 49 + 100 - 140 cos (81) = 149 - 21.9 = 127.1. Taking the square root, we get c = 11.27. So, our side measures about 11.27. When you are finding a missing side, don't forget to finish off by taking the square root to get side c by itself.
Now, let's look at an example where we find a missing angle. Remember what I said about how we can label our triangle so that it helps us to use the formula? Whatever angle we are looking for, we can label it as angle C, the side opposite it as side c, and the other two sides as side a and side b.
Find the measure of angle x.
Ah, in this picture, we have an angle x. But, our formula for the law of cosines doesn't have an x - it has a big C. What can we do? Yes, we can simply relabel our x as angle C, then our 9 as side c. Then we can arbitrarily choose between a and b for the other two sides. We will use a for 7 and b for 12. So, we have a = 7, b = 12, and c = 9. Plugging in these values into our formula, we get this:
We are going to evaluate as much as we can before solving for angle C. We get 81 = 49 + 144 - 168 cos (C). This turns into 81 = 193 - 168 cos (C). Now we can work on solving for angle C. We subtract 193 from both sides. We get -112 = -168 cos (C). Now we can divide both sides by -168. We get 0.66667 = cos (C).
Now, to get angle C by itself, we will perform the inverse cosine calculation. On your calculator, you may have to push the function or inverse button to do this. We get C = 48.2. So, our angle x measures 48.2 degrees, approximately.
What do we learn from this example? Finding the measure of an angle is a bit more complicated than finding the measure of a side. It requires a bit of algebraic manipulation of the formula to solve for angle C. We have to remember that we need to take the inverse cosine at the end to get angle C by itself. But, all in all, it wasn't too bad to work with, was it?
Let's review what we've learned now. We learned that the law of cosines is a formula to help you solve all kinds of triangles. Specifically in this video lesson, we looked at oblique triangles, triangles that are not right triangles. Our formula looks like this:
The small letters a, b, and c stand for the sides of a triangle and the big C stands for the angle opposite side c. Because we are dealing with any kind of triangle, we can label our triangle in any way that will make our problem solving easier. Try to label the side you want to find as side c or the angle that you want to find as angle C.
To use this formula to find a missing side, you will need to know the measurements of the other two sides along with the angle opposite the side you want to find. To find the measure of an angle, you will need to know the measurements of all three sides of your triangle. To find the measure of an angle, you also need to perform some algebra manipulation to solve for angle C.
You will have the ability to do the following after watching this video lesson:
- Define oblique triangle
- Identify the law of cosines
- Explain how to label a triangle when working with the law of cosines
- Use the formula to find a missing side or missing angle
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