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)

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

为什么Each 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

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

为什么b/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°

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

为什么sin 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

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

为什么Angle 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

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

为什么C = 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

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

为什么It 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

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

为什么a² = 5² + 8² − 2 × 5 × 8 × 0.5 = 25 + 64 − 40 = 49, so a = 7 cm.

120°

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

为什么The 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

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

为什么Three 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

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

为什么With 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

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

为什么The 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

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

为什么Area = 1/2 × 8 × 5 × sin 30° = 20 × 0.5 = 10 cm².

10 cm

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

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

30°

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

为什么5 = 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

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

为什么In 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.