Law of Cosines


The Law of Cosines is used to solve a triangle when Law of Sines can not be used. It can be used to find missing measures in a triangle if you know the measures of two sides and their included angle.

external image m_of_cos.gif

Equations:

For angle A:
a(a)=b(b)+c(c)-2bc(cos A)
For angle B:
b(b)=a(a)+c(c)-2ac(cos B)
For angle C:
c(c)=a(a)+b(b)-2ab(cos C)

b2=a2+c2-2ca(cosB)

Examples (From the picture above):

Problem No. 1:
Let's say that angle A is equal to 90 degrees, and angle B is equal to 40 degrees. Side c is equal to 15, b is equal to 20, and a is equal to 25. Now you want to find out what angle C is, how do you think you will do that? First, if we look at the equations, you will see that the formula for angle C is c(c)=a(a)+b(b)-2ab(cos C). All you have to do is plug in the numbers, as seen here:
15(15)=25(25)+20(20)-2(25)(20)(cos C)
Sounds confusing doesn't it? Well here's all you need to do, first, take 15(15), then 25(25), 20(20), and finally 2(25)(20). Your end product should look like this:
225=625+400-1000(cos C)
Now you can simplify this even more by adding 625 by 400:
225=1025-1000(cos C)
To simplify this even more take 225-1025:
-800=-1000(cos C)
Now you can take -800 divided by -1000:
0.8=(cos C)
Now you take arc cosine of 0.8, which should come out to be 36.86 degrees.

Law of Cosines Calculator: