Sine rule or cosine rule? How to decide

Once a triangle does not have a right angle, SOH-CAH-TOA no longer applies, and two rules take over: the sine rule and the cosine rule. Both connect the sides and angles of any triangle, but they are built for different combinations of known information, and picking the wrong one for a given question usually leads to an equation with two unknowns instead of one.

$$\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}$$

$$a^{2} = b^{2} + c^{2} - 2bc\cos A$$

A third formula from the same chapter, the area rule, is worth keeping alongside them, it finds the area of a triangle from two sides and the angle between them, without needing the height.

$$\text{Area} = \tfrac{1}{2}\,ab\sin C$$

Before reaching for either rule, one question settles which to use: is there a side and its opposite angle, both already known? If the answer is yes, the sine rule fits.

If not, the cosine rule is the one built for the gap.

In triangle $ABC$, angle $A=40°$, angle $B=65°$, and side $a=10$ cm. Find side $b$.

Angle A and its opposite side $a$ are both known, together with angle B, a complete pair plus one more angle, so the sine rule applies directly.

$$\dfrac{b}{\sin B} = \dfrac{a}{\sin A}$$

Rearranged for $b$: $b = \dfrac{a\sin B}{\sin A} = \dfrac{10\sin 65°}{\sin 40°} \approx \dfrac{10(0.9063)}{0.6428} \approx 14.1$ cm, to 3 significant figures.

In triangle $PQR$, $b=8$ cm, $c=6$ cm, and the angle between them, angle $A=60°$. Find side $a$.

Two sides and the angle between them are known, no side is paired with its own opposite angle, so the cosine rule applies.

$$a^{2} = b^{2}+c^{2}-2bc\cos A$$

Substituting: $a^{2} = 8^{2}+6^{2}-2(8)(6)\cos 60° = 64+36-96(0.5) = 100-48 = 52$, so $a=\sqrt{52}\approx7.21$ cm, to 3 significant figures.

Most errors on this topic come from the choice of rule, not from the arithmetic inside it.

• Trying the sine rule with only SAS or SSS information. Without a complete side-angle pair, the sine rule leaves an equation with two unknown ratios and cannot be solved on its own.
• Missing the ambiguous case with SSA. Stopping after the first angle found by the sine rule, without checking whether a second, supplementary angle also produces a valid triangle.
• Calculator set to the wrong angle mode. This chapter works in degrees; switching from a radians-based question earlier in the paper without resetting the mode gives a completely wrong sine or cosine value with no obvious error message.
• Rounding too early. Carrying a rounded intermediate value into the next line, rather than the full calculator value, can shift the final answer outside the accepted range.

Choosing between the sine rule and the cosine rule becomes automatic once you have sorted enough triangles by the information given rather than guessing which formula to try first. Working through a run of mixed questions with someone watching which case you reach for, and why, closes that gap faster than working alone.

Our teachers are experienced; lessons run online and are taught in English, while SPM papers themselves are set bilingually in Bahasa Melayu and English.

If solution of triangles, or trigonometry more broadly, is a sticking point, a one-hour paid trial class at the teacher's own rate (from RM50 per hour, depending on experience) is a reasonable place to start. We will not promise a particular grade, but we can help the choice between the two rules stop feeling like a guess.