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Sine & cosine rules · Lesson 1 of 2

The sine rule and the cosine rule

Two rules for triangles without a right angle: which one to use, how to set it out, and the sign of cos for obtuse angles.

20 minYou should already know: Trigonometric ratios
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  2. 2

SOH CAH TOA only works in right-angled triangles. For any other triangle there are two rules. Both use the same labels: side aa is opposite angle AA, side bb is opposite BB, and side cc is opposite CC.

ABCabc
Sides and opposite anglesSide a faces angle A, and so on
The sine ruleDrag the corners
63°47°70°a = 11b = 8.9c = 11.5ABC
12.26a ÷ sin A12.26b ÷ sin B12.26c ÷ sin C
Each side divided by the sine of the angle opposite it gives the same number. Use it when you know a side and the angle opposite it: a ÷ sin A = b ÷ sin B = c ÷ sin C.

The sine rule

ABCaba ÷ sin A = b ÷ sin B
Opposite pairsEach side goes with the angle facing it

Worked example · JAMB 1995

JAMB 1995 · UME · Q36

QRSQRS is a triangle with QS=12QS = 12 m, ∠RQS=30∘\angle RQS = 30^\circ and ∠QRS=45∘\angle QRS = 45^\circ. Calculate the length of RSRS.

  1. Match sides with opposite angles

    • RSRS is opposite ∠RQS=30∘{\angle RQS = 30^\circ}.
    • QS=12{QS = 12} is opposite ∠QRS=45∘{\angle QRS = 45^\circ}.
    • A side with its opposite angle known means the sine rule.

    Think first. Which side is opposite the 45° angle, and which is opposite the 30° angle?

  2. Write the sine rule

    RSsin⁡30∘=12sin⁡45∘\frac{RS}{\sin 30^\circ} = \frac{12}{\sin 45^\circ}

  3. Solve with exact values

    RS=12sin⁡30∘sin⁡45∘=12×1222=122=62 m\begin{aligned} RS &= \frac{12 \sin 30^\circ}{\sin 45^\circ} \\ &= \frac{12 \times \frac12}{\frac{\sqrt2}{2}} \\ &= \frac{12}{\sqrt2} = 6\sqrt2\text{ m} \end{aligned}

    The answer is C.

More: the sine rule

The cosine rule

ABCabcc² = a² + b² − 2ab cos C
Two sides and the angle betweenThe third side c is the one facing the angle C

Look at the board in cosine mode. When CC is a right angle, cos⁡C=0\cos C = 0 and the rule is just Pythagoras. The extra term corrects for the angle not being 90∘90^\circ.

More: the cosine rule

Which rule?

You knowYou wantUse
two angles and a sideanother sidesine rule
two sides and a non-included anglean anglesine rule
two sides and the angle between themthe third sidecosine rule
all three sidesan anglecosine rule

The angle bisector

The line that cuts an angle of a triangle in half divides the opposite side in the same ratio as the two sides next to the angle. In triangle MNOMNO, say the bisector of angle MM meets NONO at PP. Then

NPPO=MNMO\frac{NP}{PO} = \frac{MN}{MO}
MNOPNP : PO = MN : MO
The angle bisectorThe longer side next to M gets the longer piece

More: the angle bisector

Your turn

JAMB 1988 · UME · Q42

In triangle PQRPQR, PQ=1PQ = 1 cm, QR=2QR = 2 cm and ∠PQR=120∘\angle PQR = 120^\circ. Find the longest side of the triangle.

Worked solution (try it first)
  1. The longest side faces the largest angle, the 120∘120^\circ at QQ.
  2. So the longest side is PRPR.
  3. Cosine rule: PR2=12+22−2(1)(2)cos⁡120∘PR^2 = 1^2 + 2^2 - 2(1)(2)\cos120^\circ.
  4. cos⁡120∘=−12\cos120^\circ = -\frac12, so PR2=5+2=7PR^2 = 5 + 2 = 7 and PR=7PR = \sqrt7 cm, option D.

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