Cosine Law Examples

Once you have determined that you cannot use the sine law for a given triangle, you must use the cosine law. The cosine law works for triangles where you are given two sides and the included angle, or a triangle where the information given is all three sides. We will examine both cases here:

Example 1 - Given 2 sides and the included angle

Let's take a closer look at the triangle from the introduction page. We'll solve for side c.

Notice that we are given two sides, and the size of the angle FORMED BY these two sides. This angle is known as the 'included angle'. (It is found 'between' the two given sides). We can use the cosine law to solve for side c in this triangle. First, let's look at the cosine law:

Techinically, the cosine law is JUST ONE of the above three lines. See a pattern? The reason that we give three lines is to show how the formula can be switched around to find each side. Since we are solving for side c, we will use the third line since the 'c' variable has already been mostly isolated.

Solution

Let's fill in the information that is given. You should get:

Type the right hand side of the equation into your calculator (you should be able to do this in one long series of keystrokes). You should get:

Finally, square root both sides of the equation:

Therefore c is 39.9 cm long. Notice that now you have found the length of side c, you could revert to using the sine law to solve the remainder of the triangle, if you wished to do so.

Example 2 - Given Three Sides

Solve for angle B in the triangle below:

Solution

First we need to decide which 'line' of the cosine law we are going to use. Seeing as we are solving for angle B, we will use the line that contains angle B:

Now fill in what you know:

This is a little harder to solve than the last example. First, square all the numbers that need to be squared:

Now you can multiply the coefficients in front of the cos B:

This next step is very important. YOU CANNOT COMBINE 324 + 900 - 1080 TO GET 144! You MUST follow order of operations. The 1080 is being multiplied by the cos B, and as such CANNOT be combined with the 324 and the 900. From here, first add 324 and 900:

and subtract 1224 from both sides to get:

Now divide both sides by -1080 to isolate cos B and then solve for angle B using your calculator:

And now, seeing as you have both side b and angle B, you could use the sine law to solve for the other two angles in the triangle.

Now that you have seen the examples you can try some sample questions or head back to the cosine law introduction page.