# The Law of Cosines

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

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, and sides 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

## 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). It can be in either of these forms:

cos(C) = a2 + b2 − c2 2ab

cos(A) = b2 + c2 − a2 2bc

cos(B) = c2 + a2 − b2 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 :

 cos C = (a2 + b2 − c2)/2ab = (82 + 62 − 72)/2×8×6 = (64 + 36 − 49)/96 = 51/96 = 0.53125 C = 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:

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.