Mathematics, four units · Grade 11

When do you use the sine rule and when the cosine rule?

The data decide. Given two sides and the angle between them, or all three sides, use the cosine rule. Given a matching pair — a side and the angle opposite it — plus one more fact, use the sine rule. The short version: cosine rule when you have no complete side-and-opposite-angle pair, sine rule when you do.

Learning objectives

The rule for choosing

In a right-angled triangle, sine, cosine and tangent are enough. The moment the right angle is gone these two rules take over, and the whole skill is choosing between them.

The sine rule says the ratio of a side to the sine of its opposite angle is the same for all three sides. To use it you need a complete pair — a side with the angle facing it — plus one more piece of data.

The cosine rule generalises Pythagoras: the square of one side equals the sum of the squares of the other two, minus twice their product times the cosine of the angle between them. It takes over when there is no complete pair — that is, with two sides and the included angle, or with all three sides.

Area, and the order of operations

The area of a triangle is half the product of two sides times the sine of the angle between them. The angle must be the one enclosed by the two sides you chose — not just any angle in the triangle.

When a question asks for an area and an angle is missing, the usual route is to find the angle with the cosine rule first and then take the area with the sine. That sequence recurs on every four-unit paper.

One more shortcut: the angles of a triangle sum to 180 degrees, so the third angle is always cheaper than another calculation. Once two are known, subtract rather than apply a further rule.

The ambiguous case

When two sides and a non-included angle are given, there may be two different triangles satisfying the data. This is the ambiguous case, and it appears on the paper deliberately.

The reason: an acute angle and its supplement to 180 degrees have the same sine. A calculator always returns the acute one, and the obtuse one is no less valid.

The test: if the obtuse angle together with the given angle still totals less than 180 degrees, both triangles are possible and both must be presented. If not, there is a single solution. Stating explicitly that you checked and found only one is worth a mark.

Worked examples

  1. A triangle has sides 7 and 9 with a 60 degree angle between them. Find the third side.

    1. Two sides and the included angle — cosine rule
    2. The square: 49 plus 81 minus twice 7 times 9 times cosine 60
    3. Cosine 60 is a half, so the subtracted part is 63
    4. 130 minus 63 is 67

    Answer: The square root of 67, about 8.19

  2. In a triangle angle A is 40 degrees, the side opposite it is 10, and angle B is 65. Find the side opposite B.

    1. There is a complete pair: angle A and its opposite side — sine rule
    2. b over sine 65 equals 10 over sine 40
    3. b equals 10 times sine 65 over sine 40

    Answer: About 14.1

  3. Find the area of a triangle with sides 8 and 5 and a 30 degree angle between them.

    1. Area is half the product of the sides times the sine of the included angle
    2. Sine 30 is a half
    3. Half times 8 times 5 times a half

    Answer: 10

Common mistakes

Using Pythagoras in a triangle with no right angle
Pythagoras is the special case of the cosine rule where the angle is 90 degrees and its cosine vanishes. Without a right angle the third term is missing, and the answer is always wrong.
Taking an area with an angle that is not between the two sides
The formula needs the angle enclosed by the sides you selected. Any other angle produces a number that is not the area of any triangle.
Accepting the calculator's angle and ignoring the obtuse one
A calculator always returns the acute solution, but the sine has two solutions between zero and 180. In the ambiguous case you must test whether the obtuse one is possible and present both triangles.

What to remember

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