Sine, Cosine & Area Rules (Grade 11)

Choosing between the sine rule, the cosine rule and the area rule for non-right-angled triangles, and applying them to two-dimensional problems.

SOHCAHTOA only works in a right-angled triangle. For any other triangle you need these three rules.

1Which rule do I use?

Sine rule — when you have an angle opposite a known side (two angles and a side, or two sides and a non-included angle).
Cosine rule — when you have three sides, or two sides and the angle between them.
Area rule — when you have two sides and the angle between them and you want the area.

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Quick test: if a side and its opposite angle are both known, reach for the sine rule. If not, it is the cosine rule.

asin A = bsin B = csin C

Lower-case a is the side opposite angle A. Turn it upside down when you are solving for an angle.

In △ABC, Â = 40°, B̂ = 75° and AC = 12 units.

Worked ExampleCalculate BC in the triangle above

1
BC is opposite Â, and AC (=12) is opposite B̂.
Match each side to its opposite angle first.
2
BCsin 40° = 12sin 75°.
Sine rule.
3
BC = 12 sin 40°sin 75°.
4
= 7,99 units.
Shorter than 12, and it faces the smaller angle. Sensible.

a² = b² + c² − 2bc·cos A

and for an angle: cos A = b² + c² − a²2bc

It is Pythagoras with a correction term. If A = 90° then cos A = 0 and it collapses back to a² = b² + c².

In △ABC, AC = 8 units, AB = 5 units and Â = 60°.

Worked ExampleCalculate BC in the triangle above

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Do not work out 2bc − cos A separately. The whole term 2bc·cos A is one product — multiply all three, then subtract.

Area of △ABC = 12·ab·sin C

The two sides must sandwich the angle. Any pair works, as long as the angle sits between them.

Worked ExampleCalculate the area of the triangle with AC = 8, AB = 5 and Â = 60°

Work each one out, then click to reveal the answer.

1
In △ABC, Â = 40°, B̂ = 75° and AC = 12 units. Calculate the length of BC.
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BCsin40° = 12sin75° → BC = 7,99 units
2
In △ABC, AC = 8 units, AB = 5 units and Â = 60°. Calculate the length of BC.
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BC² = 64 + 25 − 80cos60° = 49 → BC = 7 units
3
Calculate the area of △ABC if AC = 8 units, AB = 5 units and Â = 60°.
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12(8)(5)sin60° = 17,32 square units
4
Which rule would you use given three sides and no angles?
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The cosine rule, rearranged as cos A = b² + c² − a²2bc
5
Which rule would you use given two angles and one side?
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The sine rule
6
Write down the area rule for △PQR using sides p and q.
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Area = 12·pq·sin R
7
In △ABC, a = 7, b = 8, c = 5. Calculate Â to one decimal.
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cos A = 64 + 25 − 492(8)(5) = 4080 = 0,5 → Â = 60,0°
8
Why can SOHCAHTOA not be used in △ABC above?
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It is not right-angled — SOHCAHTOA is only defined for right-angled triangles.