Law of Cosines

Hi, and welcome to this video on the law of cosines!

Hi, and welcome to this video on the law of cosines!

Let’s start by taking a look at the Pythagorean Theorem. It’s that easy-to-remember formula for finding the longest side of a right triangle when we know the length of the other two sides. Wouldn’t it be great if it worked for ALL triangles? That’s what the Law of Cosines does – it allows us to find the third length of any triangle. Here’s the formula:

The length of side c squared is equal to the length of side a squared plus the length of side b squared minus two times the length of side a times the length of side b times the cosine of angle C.

As you can see from the end of the formula we need one more piece of information than the Pythagorean Theorem requires – we need the measure of an angle. To be specific, we need the measure of the angle between the two sides we know (sides a & b).

Here’s a triangle we’ll call capital ABC and with the sides labeled lowercase a, b, and c:

As you can see, we’re using lower case letters for sides and upper case letters for angles. So in order to use the Law of Cosines to find side c, we would need to know side b, side a, and angle C, which is opposite side c.

Let’s try it out. Look at this problem:

First, we need to check to see if we have everything we need to use the Law of Cosines. We have the measure of two sides (a & b, both 10 centimeters) and the measure of the angle between those two sides (angle C, which is 45 degrees). That’s all the info we need so let’s plug a, b, and angle C into the formula:

c2 = a2 + b2 – 2ab cos(C)
c2 = (10)2 + (10)2 – 2(10)(10) cos(45)

Be careful with the order of operations here. Evaluate the exponents first, then take the cosine of 45, like this:

c2 = 100 + 100 – 200 (0.70710678)
c2 = 200 – 144.42
c2 = 55.58
c≅ 7.46 cm

When we get to the point where we have c squared equal to a number, we take the square root of each side to find that c is approximately 7.46 centimeters. A common sense check of that answer tells us that it’s reasonable since the other two sides measure 10 cm and are opposite larger angles, so our answer should be under 10, which it is.

That’s all there is to using the Law of Cosines for finding the missing side of a triangle. In fact, it’s a bit more versatile than the Pythagorean Theorem because we can label any of the sides a, b, and c as long as we label the angle opposite that side A, B, and C.

Let’s look at another problem.

We haven’t named any of the sides or angles yet. But we can see that we have been given the measure of two of the sides and the angle between them. So we should label the angle C and the sides a and b, like this:

Now we can go ahead and plug in the values:

c2 = a2 + b2 – 2ab cos(C)
c2 = (8)2 + (12)2 – 2(8)(12) cos(120)
c2 = 64 + 144 – 192 (-0.5)
c2 = 208 + 96
c2 = 304
\(\sqrt{c^2} = \sqrt{304}\)
c≅ 17.44 cm

Let’s try our common sense check again. Side c is opposite the largest angle so it should be larger than the other sides, and it is. It’s also less than the sum of the other two sides, which it must be in order for this shape to be a triangle.

You might be wondering what would happen if we had a problem with an angle that is 90° and with the two sides around it given.

Will the Law of Cosines still work? Let’s label the sides and angles:

Then we’ll plug in our values:

c2 = a2 + b2 – 2ab cos(C)
c2 = (3)2 + (4)2 – 2(3)(4) cos(90)

But watch what happens when we take the cosine of 90:

c2 = 9 + 16 – 2(3)(3) (0)

Since the cosine of 90 is zero, that whole final term is zero! So basically it just becomes the Pythagorean Theorem when angle C is 90.

c2 = 25
c = 5

Thanks for watching, and happy studying!

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by Mometrix Test Preparation | Last Updated: June 16, 2020