Extended. When you know two sides and the angle between them, or all three sides, the sine rule is stuck. The cosine rule takes over: it is Pythagoras' theorem plus a term that corrects for the angle not being 90°.
\(a^2 = b^2 + c^2 - 2bc\cos A\), where \(A\) is the angle between sides \(b\) and \(c\). To find an angle from three sides, rearrange: \(\cos A = \dfrac{b^2 + c^2 - a^2}{2bc}\). A negative cosine means an obtuse angle.
Fig. 1: two sides and the angle between them, drawn to scale (b = 7 cm, c = 9 cm, A = 50°). The cosine rule gives the third side a = 7.00 cm
Section 01
Two sides and the included angle.
In the lab, set two sides and open or close the angle between them. Watch the third side change, and watch the correction term \(2bc\cos A\) shrink to zero when the angle reaches \(90^\circ\): that is Pythagoras.
The formula: \(a^2 = b^2 + c^2 - 2bc\cos A\). The side on the left faces the angle on the right.
When \(A = 90^\circ\): \(\cos 90^\circ = 0\), and the rule becomes \(a^2 = b^2 + c^2\).
When \(A\) is obtuse: \(\cos A\) is negative, so you ADD: the third side is longer than Pythagoras would give.
Calculator: work out \(b^2 + c^2 - 2bc\cos A\) in one go, then square root. Do not subtract \(2bc\) before multiplying by \(\cos A\).
Worked example Finding the third side
In Fig. 1, \(b = 7\) cm, \(c = 9\) cm and \(A = 50^\circ\). Find \(a\).
\(A\) is between sides \(b\) and \(c\), so use the cosine rule: \(a^2 = 7^2 + 9^2 - 2 \times 7 \times 9 \times \cos 50^\circ\).
Typing \(b^2 + c^2 - 2bc\) first and then multiplying by \(\cos A\). For \(p = 8\), \(q = 11\), \(R = 72^\circ\) that gives \(\sqrt{9\cos 72^\circ} = 1.67\) instead of 11.4. The \(\cos A\) multiplies only the \(2bc\) term.
Section 02
Three sides, find an angle.
Rearranged, the cosine rule gives any angle from the three sides. Unlike the sine rule, it is never ambiguous: a negative cosine gives the obtuse angle directly.
\(\cos A = \dfrac{b^2 + c^2 - a^2}{2bc}\): the side OPPOSITE the angle you want is the one subtracted.
The largest angle is opposite the longest side; find it first if you need to check for an obtuse angle.
Worked example The largest angle
A triangle has sides 5 cm, 7 cm and 9 cm. Find its largest angle.
The largest angle \(C\) is opposite the longest side, 9 cm.
Subtracting the wrong side. The side you subtract must be the one OPPOSITE the angle you want. Subtracting 5 instead of 9 above gives the smallest angle, not the largest.
Stage 1 · Learn
Check your understanding
Four quick checks. Each correct answer earns XP and records this skill on your profile.
Quick check+10 XP
Which formula is the cosine rule for finding side \(a\)?
Quick check+10 XP
When should you use the cosine rule to find a side?
Quick check+10 XP
In triangle \(ABC\), \(b = 3\), \(c = 5\) and \(A = 60^\circ\). Find \(a\).
Quick check+10 XP
Using the cosine rule, you find \(\cos C = -0.35\). What does this tell you about angle \(C\)?
Section 03
Sine rule or cosine rule?
Look at what you know. Try the 'which rule?' tab of the lab to practise the decision before you calculate.
You know
Use
A side and its opposite angle (a complete pair), plus one more side or angle
Sine rule
Two sides and the angle BETWEEN them
Cosine rule for the third side
All three sides
Cosine rule for any angle
A right angle
Pythagoras and SOH CAH TOA are quicker
Worked example A journey with a turn
A boat sails 12 km on a bearing of \(050^\circ\), then 8 km on a bearing of \(170^\circ\). How far is it from its starting point?
At the turning point, the bearing back to the start is \(050^\circ + 180^\circ = 230^\circ\).
The angle inside the triangle at the turn is \(230^\circ - 170^\circ = 60^\circ\), between the two known sides: cosine rule.