The Law of Cosines

The Law of Cosines (also called the Cosine Rule) is very useful for solving triangles:

Law of Cosines

It works for any triangle:

a, b and c are sides.

C is the angle opposite side c

Let's see how to use it in an example:

Example: How long is side "c" ... ?

We know angle C = 37º, a = 8 and b = 11

The Law of Cosines says:   c2 = a2 + b2 − 2ab cos(C)
     
Put in the values we know:   c2 = 82 + 112 − 2 × 8 × 11 × cos(37º)
     
Do some calculations:   c2 = 64 + 121 − 176 × 0.798…
     
More calculations:   c2 = 44.44...
     
Take the square root:   c = √44.44 = 6.67 to 2 decimal places


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:

Pythagoras Theorem:   a2 + b2 = c2   (only for Right-Angled Triangles)
         
Law of Cosines:   a2 + b2 − 2ab cos(C) = c2   (for all triangles)

So, to remember it:

  • think "abc": a2 + b2 = c2,
  • then a 2nd "abc": 2ab cos(C),
  • and put them together: a2 + b2 − 2ab cos(C) = c2

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:

Start with:   c2 = a2 + b2 − 2ab cos(C)
     
Put in a, b and c   82 = 92 + 52 − 2 × 9 × 5 × cos(C)
     
Calculate:   64 = 81 + 25 − 90 × cos(C)
     
Now we use our algebra skills to rearrange and solve:
     
Subtract 25 from both sides:   39 = 81 − 90 × cos(C)
Subtract 81 from both sides:   −42 = −90 × cos(C)
Swap sides:   −90 × cos(C) = −42
Divide both sides by −90:   cos(C) = 42/90
Inverse cosine:   C = cos-1(42/90)
     
Calculator:   C = 62.2° (to 1 decimal place)

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 c2 = a2 + b2 − 2ab cos(C) formula):

law of cosines alt

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 :

cos C  = (a2 + b2 − c2)/2ab
  = (82 + 62 − 72)/2×8×6
  = (64 + 36 − 49)/96
  = 51/96
  = 0.53125
   
= cos-1(0.53125)
  = 57.9° to one decimal place

 

Versions for a, b and c

Also, you can rewrite the c2 = a2 + b2 - 2ab cos(C) formula into "a2=" and "b2=" form.

Here are all three:

Law of Cosines

Law of Cosines

Law of Cosines

But it is easier to remember the "c2=" 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

Start with:   c2 = a2 + b2 − 2ab cos(C)
     
x for a, y for b and z for c   z2 = x2 + y2 − 2xy cos(Z)
     
Put in the values we know:   z2 = 9.42 + 6.52 − 2×9.4×6.5×cos(131º)
Calculate:   z2 = 88.36 + 42.25 − 122.2 × (−0.656...)
    z2 = 130.61 + 80.17...
    z2 = 210.78...
    z = √210.78... = 14.5 to 1 decimal place.

 

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).