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:

c² = a² + b² − 2ab cos(C)

It helps us solve some triangles. Let's see how to use it.

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

trig cos rule example

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

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:
(only for Right-Angled Triangles)a² + b² = c²

Law of Cosines:
(for all triangles)a² + b² − 2ab cos(C) = c²

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" ...?

trig cos rule example

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)

triangle SSS

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, we can rewrite the c² = a² + b² − 2ab cos(C) formula into a²= and b²= form.

Here are all three:

a² = b² + c² − 2bc cos(A)

b² = a² + c² − 2ac cos(B)

c² = a² + b² − 2ab cos(C)

But it is easier to remember the "c²=" form and change the letters as needed !

As in this example:

Example: Find the distance "z"

trig cos rule example

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 cz² = 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).