Geometry and measures, Year 11: the sine rule, the cosine rule and triangle area

a/sin A = b/sin B = c/sin C (or the same upside down: sin A/a = sin B/b = sin C/c)

HintEach side is paired with something directly across the triangle from it, and every pair gives an equal result.

The whyEach side is divided by the sine of the angle opposite it, and all three results are equal. It can also be written the other way up, sin A/a = sin B/b = sin C/c, which is handier when finding an angle. The specification says this rule is to be known and applied.

12.3 cm

HintWrite side over sine of its opposite angle for both pairs and set the two fractions equal.

The whyb/sin 50° = 8/sin 30°, so b = 8 × sin 50° ÷ sin 30° = 8 × 0.7660 ÷ 0.5 = 12.26, which is 12.3 cm to one decimal place.

44.4°

HintPut the sines on top this time, find sin B as a decimal first, then work back to the angle.

The whysin B/14 = sin 30°/10, so sin B = 14 × 0.5 ÷ 10 = 0.7. Then B = sin⁻¹(0.7) = 44.4° to one decimal place.

The sine rule needs a side and its opposite angle as a known pair, and there is no such pair here

HintMatch each given measurement to whatever lies across the triangle. Do you hold both halves of any match?

The whyAngle P is opposite QR, which is unknown; PQ is opposite angle R and PR is opposite angle Q, both unknown. Two sides and the angle between them is the case for the cosine rule instead.

65; 7.1

HintThe angles of any triangle share out 180°. Once the third one is found, a matched pair is available.

The whyC = 180 − 40 − 75 = 65°. Then a/sin 40° = 10/sin 65°, so a = 10 × sin 40° ÷ sin 65° = 7.09, which is 7.1 cm.

a² = b² + c² − 2bc cos A

HintStart as Pythagoras' theorem does, then take away a correction built from the other two sides and the angle facing side a.

The whyIt links all three sides with one angle. Use it to find the third side from two sides and the angle between them, or rearranged as cos A = (b² + c² − a²) ÷ (2bc) to find an angle from three sides. The specification says this rule is to be known and applied.

7

HintSquare the two sides and add, then take away twice their product times the cosine. Remember the square root at the end.

The whya² = 5² + 8² − 2 × 5 × 8 × 0.5 = 25 + 64 − 40 = 49, so a = 7 cm.

120°

HintThe largest angle faces the longest side, so that side plays the part of a in the rule.

The whyThe largest angle is opposite the 7 cm side. cos A = (3² + 5² − 7²) ÷ (2 × 3 × 5) = (9 + 25 − 49) ÷ 30 = −0.5, so A = cos⁻¹(−0.5) = 120°. A negative cosine tells you the angle is obtuse.

The cosine rule; the sine rule needs at least one known angle paired with its opposite side

HintOne of the two formulas contains two angles, so it cannot start when you have none.

The whyThree sides, or two sides and the angle between them, call for the cosine rule. The sine rule is used when a side and its opposite angle are both known, together with one more side or angle.

Pythagoras' theorem

HintWith the correction gone, only three squares remain, linked by an equals sign and a plus sign.

The whyWith cos 90° = 0 the term 2bc cos A disappears and a² = b² + c² is left, with a as the hypotenuse. So the cosine rule is Pythagoras' theorem with a correction for angles that are not right angles.

between

HintThink about where angle C has to sit compared with the two sides in the formula.

The whyThe formula needs two sides and the included angle, the one they form where they meet. With sides a and b that angle is C, because angle C is opposite side c. The specification says this formula is to be known.

10

HintMultiply the two sides, halve, and then multiply by the sine of the angle they form.

The whyArea = 1/2 × 8 × 5 × sin 30° = 20 × 0.5 = 10 cm².

10 cm

HintWrite the area formula with a letter for the missing side, then solve the equation.

The why15 = 1/2 × 6 × b × 0.5 = 1.5b, so b = 15 ÷ 1.5 = 10 cm.

30°

HintPut everything into the area formula and solve for the sine of the angle first.

The why5 = 1/2 × 4 × 5 × sin C = 10 sin C, so sin C = 0.5 and C = 30°. (An angle of 150° has the same sine, which is why the question says the angle is acute.)

Height = b sin C

HintDrop a perpendicular from the top corner to the base; side b becomes the hypotenuse of a right-angled triangle.

The whyIn the small right-angled triangle, the height is opposite angle C and b is the hypotenuse, so sin C = height ÷ b and height = b sin C. Then area = 1/2 × base × height = 1/2 × a × b sin C.