After watching this video lesson, you will be able to use the law of cosines to solve real-world problems. Learn what kinds of real-world problems the law of cosines can help you solve.

The Law of Cosines

Trigonometry has many useful formulas. In this video lesson, we are going to discuss the law of cosines. This useful formula is used to solve triangles of all kinds. As you know, our trigonometric functions of sine, cosine, and tangent have their beginnings in a right triangle.

Well, delving deeper into trigonometry and its functions, we are able to use our trigonometric functions to help us solve not just right triangles but other kinds of triangles as well. It opens up a whole new door for us. Are you ready to dig into this law of cosines? Okay! Let’s go. We begin with the formula:

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law of cosines

Do you see any resemblance with another popular formula that you deal with on a fairly regular basis? That’s right, the Pythagorean theorem: a^2 + b^2 = c^2. There are some differences here though, but seeing the connection between this law of cosines and the Pythagorean theorem will help you remember it.

Once you have the a, b, and c down, what is left is the 2ab cos(C). This can be easy to remember as well, if you notice that you have all three letters in order. We have a, b, and then c. And because this is the second set of a, b, and c, we have a number two in front. Instead of adding this part, we subtract it. Do your best to remember this formula. Can you think of other memory aids that will help you?

One more difference is that a, b, and c in the law of cosines all refer to different sides of a triangle. There’s no hypotenuse anymore since we are dealing with triangles of all kinds, not just right triangles. The big C inside the cosine argument stands for the angle opposite side c:

law of cosines

Real-World Uses

This law of cosines is very useful in the real world where triangles are involved. Where are triangles involved in the real world? Surveyors use triangles to calculate an unknown distance where they know the other two distances of the triangle.

The beauty of the law of cosines can be seen when you want to find the location of a fire, for example. You see the fire in the distance, but you don’t know how far away it is. However, to the left of the fire is a fast food restaurant that you know is exactly 1 mile away.

A friend of yours standing somewhere else tells you that the fire is 2 miles away from the fast food restaurant. Your friend also tells you that the angle at the fast food restaurant that is formed by where you are standing and where the fire is equals 54 degrees.

Because you know two sides of the triangle, along with the angle that is opposite your unknown side, you can use the law of cosines to help you calculate that third side:

law of cosines

Example 1

Let’s go ahead and solve this problem as our first example. We know two sides of the triangle. One measures 1 mile and the other measures 2 miles. The third side is the side we are looking for. The angle opposite this side is 54 degrees.

We can use our law of cosines to help us solve this. Because this is not a special triangle like a right triangle, we can arbitrarily label our sides. Since we are looking for our third side, we will label our third side as c. We can label the other two sides with a and b.

It doesn’t matter which side is which as long as we keep the sides straight after we’ve labeled them. We will label the 1 mile side as a and the 2 mile side as b. Our angle C is then 54. Now, we can plug all these numbers into our formula for the law of cosines:

law of cosines

All that is left to do now is to evaluate our formula to find our answer. We get c^2 = 1 + 4 – 4 * cos(54) = 5 – 2.35 = 2.65. Taking the square root, we get c = 1.63 miles. So the fire is 1.63 miles away from where you are.

Example 2

Let’s look at another example. You are standing 10 feet away from a tree that is 30 feet tall. You need to figure out how long of a ladder you need to go from where you are to the top of the tree.

This tree happens to be a fruit tree, a plum tree to be exact. And at the very top of this tree, there is a giant plum just ready for the picking. The angle the base of the tree makes between you and the top of the tree is 90 degrees.

To figure out how long of a ladder you need, we can again use our law of cosines. We can label the 10 feet as a and the 30 feet as b. Our c is the length of the ladder. Our big C is 90. Plugging these values into our formula, we get this:

law of cosines

Evaluating this, we get c^2 = 100 + 900 – 600 * cos(90) = 100 + 900 – 600 * 0 = 1000. Taking the square root, we get c = 31.62 feet. So, we need a ladder that is 31.62 feet long.

Lesson Summary

Let’s review what we’ve learned. The law of cosines is a useful formula that is used to solve triangles of all kinds. The formula for the law of cosines is c^2 = a^2 + b^2 – 2ab cos(C), where a, b, and c are the sides of the triangle and the big C is the angle opposite the c side.

The law of cosines is used in the real world by surveyors to find the missing side of a triangle, where the other two sides are known and the angle opposite the unknown side is known. The law of cosines is also used whenever a triangle is involved.

To use the law of cosines formula, we simply plug in our two known sides into a and b, and then our angle into C. To find the answer, we evaluate the formula to find our c.

Learning Outcomes

Studying this video lesson might help you realize the following goals:

  • State the purpose of the law of cosines
  • Use the law of cosines to solve for the missing length of a triangle