MISSION · Geometry
The Sine and Cosine Rules
Trigonometry for the other triangles
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The idea in one sentence
SOH-CAH-TOA needs a right angle; these two rules work on any triangle at all, and between them they solve every case.
The sine rule
\[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\]Each side is paired with the angle opposite it — side \(a\) with angle \(A\) — and all three ratios are equal. In words: the bigger the angle, the longer the side facing it, and in exactly this proportion.
Use it when you have a matched pair: a side and its opposite angle, plus one more piece of information. That covers two situations:
- Two angles and any side (the third angle is free, since they total \(180°\)).
- Two sides and an angle opposite one of them.
You only ever need two of the three ratios at a time. Pick the pair that contains what you know and what you want, then cross-multiply.
The cosine rule
\[c^2 = a^2 + b^2 - 2ab\cos C\]Use it when the sine rule can’t start, which is exactly when you have no matched pair:
- Two sides and the angle between them.
- All three sides, and you want an angle.
Look at what happens when \(C = 90°\): \(\cos 90° = 0\), the last term vanishes, and you’re left with \(c^2 = a^2 + b^2\). The cosine rule is Pythagoras with a correction term for the angle not being right. If \(C\) is acute the correction subtracts, making \(c\) shorter; if obtuse, \(\cos C\) is negative and the term adds, making \(c\) longer. That’s the whole meaning of the formula.
To find an angle instead, rearrange:
\[\cos C = \frac{a^2 + b^2 - c^2}{2ab}\]Choosing between them
Count what you have. A matched side-and-opposite-angle pair means the sine rule, which is less arithmetic. No matched pair means the cosine rule, which always works but takes longer. Start with the cosine rule to break into the triangle, then switch to the sine rule for anything still missing.
The ambiguous case
One warning. Given two sides and a non-included angle, there may be two different triangles that fit — one acute and one obtuse.
The cause is that \(\sin 30°\) and \(\sin 150°\) are both 0.5, so the inverse sine on your calculator returns only one of two valid angles. It hands back the acute one and stays quiet about the other.
Whenever you use the sine rule to find an angle, check whether \(180° -\) your answer also fits — it does whenever the remaining angles still total less than \(180°\). Finding a side is safe; finding an angle needs the check.
Common mix-ups
- Pairing a side with the wrong angle. The angle must be opposite the side.
- Using the cosine rule with a non-included angle. The angle has to sit between the two known sides.
- Missing the second solution in the ambiguous case.
- Rounding mid-calculation. Keep full precision until the final answer, or the error compounds badly in these formulas.
Check yourself
A triangle has sides of 7 and 9 with an angle of \(60°\) between them. Find the third side.
No matched pair, so the cosine rule: \(c^2 = 49 + 81 - 2(7)(9)(0.5) = 67\), giving \(c \approx 8.19\).
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