POP

E6.5 The Sine Rule, Cosine Rule & Area of Any Triangle

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01 — Explanation

Why doesn't SOHCAHTOA work here?

SOHCAHTOA needs a right angle. In triangle $ABC$ with no right angle (side $a$ opposite $A$ , $b$ opposite $B$ , $c$ opposite $C$ ), three different tools take over: the sine rule, the cosine rule, and a trig-based area formula. All three, along with the quadratic formula, are printed in the exam's List of Formulas — the skill being tested is choosing and applying the right one, not memorising it.

02 — Explanation

How do I decide which rule to use?

Given	Use
2 angles + 1 side (AAS/ASA)	Sine rule
2 sides + a non-included angle (SSA)	Sine rule — watch for the ambiguous case
2 sides + the included angle (SAS)	Cosine rule for the side, then the area formula
All 3 sides (SSS)	Cosine rule for an angle

Labelling the triangle first — writing $a$ , $b$ , $c$ opposite their matching capital-letter angles — makes it far easier to see which row of this table applies.

03 — Explanation

How do I use the sine rule?

\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C} \ \text{(side)} \qquad \frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c} \ \text{(angle)}

Only ever work with one pair of fractions from the three available at a time — pick the pair where three of the four values are already known.

04 — Explanation

What is the "ambiguous case"?

When the sine rule is used to find an angle from an SSA setup, $\sin(180°-A)=\sin A$ means two triangles can fit the data. With $a=9$ , $b=12$ , $A=38°$ :

\sin B = \frac{12\sin(38°)}{9} = 0.8209... \implies B_1 = \sin^{-1}(0.8209) = 55.2°, \quad B_2 = 180°-55.2° = 124.8°

Both are valid here since $A+B_1=93.2°<180°$ and $A+B_2=162.8°<180°$ .

Examiner Tip

Whenever a question gives two sides and an angle that is not between them, assume there might be two answers until you've checked — omitting the obtuse solution is the most common way this topic loses marks.

05 — Explanation

How do I use the cosine rule?

a^2=b^2+c^2-2bc\cos A \ \text{(side)} \qquad \cos A = \frac{b^2+c^2-a^2}{2bc} \ \text{(angle)}

With $a=7$ , $b=9$ , $c=10$ : $\cos A = \dfrac{81+100-49}{180} = \dfrac{132}{180}$ , so $A = \cos^{-1}(0.7\overline3) = 42.8°$ (1 d.p.). Treat $2bc\cos A$ as one block — evaluate $b^2+c^2$ first, then subtract that whole product, rather than splitting it apart.

06 — Explanation

How do I find the area without knowing the height?

\text{Area} = \frac12 ab\sin C

where $C$ must be the angle trapped between the two sides $a$ and $b$ being used — not an angle opposite either of them. Using the wrong angle (one that isn't included between the two given sides) is a common way marks are lost even when the formula itself is written correctly.

07 — Worked example

Worked Example: a three-part exam-style question

In triangle $PQR$ , $PQ=14$ cm, $PR=11$ cm, and the included angle $\angle QPR = 42°$ .

(a) Find the area of the triangle, correct to 3 s.f.

(b) Find the length $QR$ , correct to 3 s.f.

(c) Find $\angle PQR$ , correct to 1 d.p.

Solution

(a) The angle $42°$ is included between the two given sides, so the area formula applies directly:

\text{Area} = \frac12(14)(11)\sin(42°) = 77 \times 0.669130... = 51.5 \text{ cm}^2 \ (3\text{ s.f.})

(b) $QR$ is opposite the known angle $P$ , and both adjacent sides are known — this is SAS, so the cosine rule finds the missing side directly:

QR^2 = 14^2+11^2-2(14)(11)\cos(42°) = 196+121-308(0.74314...) = 317-228.89 = 88.11

QR = \sqrt{88.11} = 9.39 \text{ cm } (3\text{ s.f.})

(c) With all three sides now available (using the unrounded $QR=9.3866...$ ), the sine rule finds the remaining angle. $\angle PQR$ is opposite side $PR=11$ :

\frac{\sin(\angle PQR)}{11} = \frac{\sin(42°)}{9.3866} \implies \sin(\angle PQR) = \frac{11\sin(42°)}{9.3866} = 0.78420...

\angle PQR = \sin^{-1}(0.78420) = 51.6° \ (1\text{ d.p.})

Since $\angle PQR$ is clearly the smallest-looking angle in a triangle where the largest given angle is $42°$ , and $42°+51.6°=93.6°<180°$ , there's no need to check the obtuse alternative here — the ambiguous case only needs to be considered when the sine rule is used to find an angle from an SSA (not SAS-derived) starting setup.