The Law of Cosines
For any triangle:
a, b and c are sides. C is the angle opposite side c |
The Law of Cosines (also called the Cosine Rule) says:
c^{2} = a^{2} + b^{2} − 2ab cos(C)
It helps us solve some triangles. Let's see how to use it.
Example: How long is side "c" ... ?
We know angle C = 37º, and sides a = 8 and b = 11
Answer: c = 6.67
How to Remember
How can you remember the formula?
Well, it helps to know it's the Pythagoras Theorem with something extra so it works for all triangles:
(only for Right-Angled Triangles)a^{2} + b^{2} = c^{2}
(for all triangles)a^{2} + b^{2} − 2ab cos(C) = c^{2}
So, to remember it:
- think "abc": a^{2} + b^{2} = c^{2},
- then a 2nd "abc": 2ab cos(C),
- and put them together: a^{2} + b^{2} − 2ab cos(C) = c^{2}
When to Use
The Law of Cosines is useful for finding:
- the third side of a triangle when we know two sides and the angle between them (like the example above)
- the angles of a triangle when we know all three sides (as in the following example)
Example: What is Angle "C" ...?
The side of length "8" is opposite angle C, so it is side c. The other two sides are a and b.
Now let us put what we know into The Law of Cosines:
Now we use our algebra skills to rearrange and solve:
In Other Forms
Easier Version For Angles
We just saw how to find an angle when we know three sides. It took quite a few steps, so it is easier to use the "direct" formula (which is just a rearrangement of the c^{2} = a^{2} + b^{2} − 2ab cos(C) formula). It can be in either of these forms:
cos(C) = \frac{a^{2} + b^{2} − c^{2}}{2ab}
cos(A) = \frac{b^{2} + c^{2} − a^{2}}{2bc}
cos(B) = \frac{c^{2} + a^{2} − b^{2}}{2ca}
Example: Find Angle "C" Using The Law of Cosines (angle version)
In this triangle we know the three sides:
- a = 8,
- b = 6 and
- c = 7.
Use The Law of Cosines (angle version) to find angle C :
Versions for a, b and c
Also, we can rewrite the c^{2} = a^{2} + b^{2} − 2ab cos(C) formula into a^{2}= and b^{2}= form.
Here are all three:
a^{2} = b^{2} + c^{2} − 2bc cos(A)
b^{2} = a^{2} + c^{2} − 2ac cos(B)
c^{2} = a^{2} + b^{2} − 2ab cos(C)
But it is easier to remember the "c^{2}=" form and change the letters as needed !
As in this example:
Example: Find the distance "z"
The letters are different! But that doesn't matter. We can easily substitute x for a, y for b and z for c
Answer: z = 14.5
Did you notice that cos(131º) is negative and this changes the last sign in the calculation to + (plus)? The cosine of an obtuse angle is always negative (see Unit Circle).