Try each one on paper first, then check a single step. That is worth far more than reading a finished solution.
Example 1 — the sine rule for a side
In triangle ABC, A=40°, B=75° and a=8 cm. Find b.
- 1You have side a with its opposite angle A — a matching pair — so the sine rule applies.Check for a matching pair before anything else.
- 2sin75°b=sin40°8Keep the unknown on top.
- 3b=sin40°8sin75°Multiply both sides by sin75°.
- 4b=12.0 cm (3 s.f.)Sensible: B is the larger angle, so b must be the longer side.
Example 2 — the cosine rule for an angle
A triangle has sides 5 cm, 6 cm and 7 cm. Find its largest angle.
- 1There is no matching side-and-opposite-angle pair, so the sine rule cannot start.Three sides given means the cosine rule.
- 2The largest angle faces the longest side, so take a=7, b=5, c=6.
- 3cosA=2bcb2+c2−a2=2(5)(6)25+36−49
- 4=6012=0.2A positive cosine means an acute angle.
- 5A=cos−1(0.2)=78.5° (1 d.p.)Had the cosine come out negative, the angle would simply be obtuse.
Example 3 — area from two sides and the angle between
Two sides of a triangle are 6 cm and 8 cm, with an angle of 50° between them. Find the area.
- 1The angle lies between the two sides, so the formula applies.It must be the included angle.
- 2Area =21absinC
- 3=21(6)(8)sin50°
- 4=24sin50°Do the halving and multiplying first — it is easier to check.
- 5=18.4 cm2 (3 s.f.)Less than 21(6)(8)=24, which is the area when the angle is 90° ✓