Three mobile masts outside Khon Kaen sit 5 km, 6 km and 8 km apart. Find the angle at the mast opposite the 8 km side. There is a right way and a way that gives 87.1°, and the screen looks the same either way.
The answer is 92.9°. If you got 87.1° you used the sine rule, and the sine rule cannot tell an obtuse angle from an acute one. Here is why, on one picture.
One height, two angles. The inverse sine key has to pick one, and it always picks the one below 90°. Where a triangle settles it, the figure marks the one you want with a tick.
Sides 5, 6 and 8. The sine rule route starts by finding a different angle, then chains:
The angle sum is the check. The three angles total 174.27°, which is 5.7° short of 180, and that shortfall is the only sign anything went wrong. Add your three angles up. It costs two seconds and it catches this every time.
Add the rounded values instead, 38.6 + 48.5 + 87.1, and you get 174.2. Rounding three numbers before adding them moves the total, which is its own reason to keep full accuracy until the last line.
The cosine rule asks for the same angle directly and cannot make the mistake:
cos C = (5² + 6² − 8²) / (2 × 5 × 6) = −3/60 = −0.05
C = cos⁻¹(−0.05) = 92.9°
The minus sign is the whole point. Cosine is negative above 90° and positive below it, so the sign of your cosine tells you which side of a right angle you are on. Sine is positive on both sides and tells you nothing.
If you are not convinced 87.1° is wrong, build the triangle from it. Two sides of 5 and 6 with 87.1° between them close on a third side of 7.61, not 8. The triangle does not reach.
You never choose between the two rules. What you were given chooses for you, and only one of the five cases is even allowed to be ambiguous.
| What you are given | What starts |
|---|---|
| A right angle anywhere in it | sin, cos, tan. Do not reach for a rule. |
| Two angles and any side | Sine rule. The third angle is free from the angle sum. |
| Two sides and the angle between them | Cosine rule, for the third side. |
| All three sides | Cosine rule, for any angle you like. |
| Two sides and an angle not between them | Sine rule. It is set where it has one answer; the ambiguous version of it is what the course leaves out. |
The habit that removes the problem. When you have three sides, find the largest angle first, with the cosine rule. A triangle can hold at most one obtuse angle, so once the big one is settled every remaining angle is acute and the sine rule is safe for the rest.
Sine rule, for sides and angles in facing pairs:
a / sin A = b / sin B = c / sin C
Cosine rule, for a side:
c² = a² + b² − 2ab cos C
Rearranged, for an angle:
cos C = (a² + b² − c²) / 2ab
Area:
Area = ½ ab sin C, where C is the angle between a and b
½ ab sin C needs the included angle. Any other angle gives a number, and the number is wrong. For the masts, the area is ½ × 5 × 6 × sin 92.9° = 14.98 km², and 5 and 6 are the two sides either side of 92.9°.
A worked one with the cosine rule forwards. Sides 7 and 9 with 52° between them:
There is no sine-rule or cosine-rule key. You type the rearranged formula, which means the only thing that can go wrong is the mode and the brackets.
When you may use it. Applications allows a calculator in every paper. Analysis does not allow one in Paper 1, so if you sit AA you must be able to run both rules by hand as well.
The mark people lose. c has to be the side opposite the angle you want. Pair them wrongly and you get (5²+8²-6²)/(2×5×8) = 0.6625, so 48.5°, and nothing complains: 48.5° really is an angle of that triangle, just not the one you were asked for. That is why it survives checking. Two smaller ones: brackets round the whole denominator, or the division happens before the multiplication and the machine returns a domain error; and radians, where cos⁻¹(−0.05) comes back as 1.62.
1. A triangle has sides 7 and 9 with 52° between them. Find the third side, to 2 decimal places.
2. A triangle has sides 5, 6 and 8. Find its largest angle, to 1 decimal place.
3. A student finds all three angles of a triangle and gets 174.27° in total. What is the single most likely cause?
4. Find the area of the mast triangle: sides 5 and 6 with 92.9° between them, to 2 decimal places.
1 markThe correct rule, written down with the right letters in the right places.
1 markSubstituting correctly, including the included angle where the formula needs one.
1 markThe answer, to the accuracy asked for.
An obtuse angle found with the sine rule loses the final mark and often the ones that depend on it. Finding the largest angle with the cosine rule first is not a tip, it is how the method is meant to go.
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