Cosine rule

#### Tier: #Higher

Description

The cosine rule applies to any triangle. Use it when you know:

• All three sides (to find an angle), or
• Two sides and the included angle (to find the third side)

Finding a side: $$a^2 = b^2 + c^2 - 2bc\cos A$$

Finding an angle (rearranged): $$\cos A = \frac{b^2 + c^2 - a^2}{2bc}$$

Example — finding a side: $b = 5$, $c = 8$, $A = 60°$: $$a^2 = 25 + 64 - 2(5)(8)\cos 60° = 89 - 80 \times 0.5 = 89 - 40 = 49$$ $$a = 7$$

Notice that when $A = 90°$, $\cos 90° = 0$, so the cosine rule reduces to Pythagoras.

Memory aid: same letter pairing — side $a$ opposite angle $A$, side $b$ opposite $B$, etc.

Common error: substituting into the wrong version of the formula (use the rearranged version to find angles, not the first version).

Links

Substitution Sine rule Bearings Pythagoras

Use SHIFT cos⁻¹ to find angle from $\cos A$ value. Take care with negative cosines — obtuse angles have negative cosines.