The cosine rule

The cosine rules are, like the sine rule, a way we can find angles and side lengths in any triangle. There are three cosine rules, one for each of the sides $a$, $b$ and $c$. Picking the right one of the three is the only real decision. After that it's inserting numbers and, when it's an angle you want, one extra step at the end.

When do I use this?

The cosine rules can be used if we either have three side lengths in the triangle and want to find some angles, or if we have an angle and the two side lengths that don't belong to the angle, that is, the two sides next to it. If instead you know an angle together with its own opposite side, it's the sine rule you want. And if the triangle has a right angle, the short formulas from sin, cos and tan in right-angled triangles are enough.

The three cosine rules

$$a^2 = b^2 + c^2 - 2bc \times \cos(A)$$

$$b^2 = a^2 + c^2 - 2ac \times \cos(B)$$

$$c^2 = a^2 + b^2 - 2ab \times \cos(C)$$

The naming is the usual one: side $a$ is opposite corner $A$, side $b$ opposite corner $B$, side $c$ opposite corner $C$. Notice that each rule contains exactly one angle, and it's the angle opposite the side on the left. That's how you pick: the version that contains the angle you know or want.

Worked example: finding a side

Let's look at an example:

If we have the quantities as shown on the figure, and want to find the side length $a$, we can use the first cosine rule, because that's the one with angle $A$ in it:

$$a^2 = b^2 + c^2 - 2bc \times \cos(A)$$

We insert our quantities in the formula. The side $b$ is opposite $B$, so it's the side of length $6$, and $c$ is the side of length $4$:

$$a^2 = 6^2 + 4^2 - 2 \times 6 \times 4 \times \cos(35^\circ)$$

That gives $a^2 \approx 12.68$, and since we want $a$ and not $a^2$, we take the square root:

$$a = \sqrt{12.68} \approx 3.56$$

So the side $a$ is about $3.56$.

Worked example: finding an angle

We could also find an angle, if we had been given three lengths:

If we want to find angle $B$ on the figure above, we have to use the cosine rule that contains angle $B$. We can see that the second cosine rule contains angle $B$:

$$b^2 = a^2 + c^2 - 2ac \times \cos(B)$$

So we just insert the side lengths in their places in the cosine rule. Side $b$ is opposite $B$, so $b = 5$, and the other two are $a = 3$ and $c = 7$:

$$5^2 = 3^2 + 7^2 - 2 \times 3 \times 7 \times \cos(B)$$

Now we have to solve this equation to find $B$. We move everything except $\cos(B)$ over to the other side, which gives

$$\frac{3^2 + 7^2 - 5^2}{2 \times 3 \times 7} = \cos(B)$$

Now, to get $B$ completely on its own, we take the opposite of cos on both sides, which is $\cos^{-1}$:

$$\cos^{-1}\left(\frac{3^2 + 7^2 - 5^2}{2 \times 3 \times 7}\right) = B$$

(If this $\cos^{-1}$ doesn't quite make sense yet, it doesn't matter. It's the button that undoes cos, and you can read more about that kind of button in the guide on inverse functions.)

The fraction works out to $\frac{3^2 + 7^2 - 5^2}{2 \times 3 \times 7} = \frac{33}{42}$, and the calculator gives

$$B \approx 38.21^\circ$$

So angle $B$ is about $38.21$ degrees.

Common mistakes

• Picking a version that doesn't contain your angle. Each cosine rule has exactly one angle in it, the one opposite the side on the left. To find or use angle $B$, you need the version that starts with $b^2$.
• Putting the sides in the wrong places. The side on the left is the one opposite the angle in the rule. In the angle example, $b = 5$ because it's opposite $B$, and the other two sides fill $a$ and $c$.
• Using the cosine rule where the sine rule is simpler. If you know an angle and its own opposite side, the sine rule gets there in one line. The cosine rule is for three sides, or two sides and the angle between them.
• Stopping at cos(B). $\cos(B) = \frac{33}{42}$ is not the angle. You still have to take $\cos^{-1}$ on both sides to get $B$ itself.

Related

• Trigonometry: sine, cosine and tangent
• The sine rule, when you have an angle and its opposite side
• Sin, cos and tan in right-angled triangles
• Inverse functions, what the $\cos^{-1}$ button is