Sine & Cosine Rule

Solving any triangle — not just right-angled ones. Plus the ambiguous case. ✎

① Labelling a Triangle

lower case

Get this right first — every formula on this sheet depends on the side sitting opposite its angle.

② Sine Rule ✦ formula booklet

a / sin A = b / sin B = c / sin C

Use it when you have a matching pair:

2 angles + 1 side (AAS) → find a side.

2 sides + 1 opposite angle (SSA) → find an angle.

Looking for an angle? Flip it over — use sin A / a = sin B / b to keep the unknown on top.

③ Cosine Rule ✦ formula booklet

to find a SIDE

a² = b² + c² − 2bc · cos A

to find an ANGLE

cos A = b² + c² − a² / 2bc

Use it when the sine rule can't start:

2 sides + the angle BETWEEN them (SAS) → find the 3rd side.

All 3 sides (SSS) → find any angle.

−2bc cos A

b², c²

④ Which Rule? — Decision Table

SSS — 3 sides

Cosine (angle form)

SAS — angle between

Cosine (side form)

SSA — non-included

Sine rule (ambiguous!)

right angle + 2 things

SOH CAH TOA

Quick test: can you see a side and its opposite angle both known? → sine rule. If not → cosine rule.

⑤ The Ambiguous Case (SSA)

When you use the sine rule to find an angle, there can be two valid answers.

same a, same c, same A — two different triangles

if θ works, so does 180° − θ

(because sin θ = sin(180° − θ))

Your GDC only ever gives the acute answer.

Check it: add your angle to the known one. If the total is < 180°, that option is valid.

Only happens when finding an angle with the sine rule — never with the cosine rule.

★ Worked example ✎

In triangle ABC, b = 7 cm, c = 9 cm and A = 62°. Find a, then find angle B.

Spot the type:

a² = 7² + 9² − 2(7)(9)cos 62° = 49 + 81 − 126 cos 62° ≈ 130 − 59.15 = 70.85

a = 70.85 ≈ 8.42 cm

Check: 62 + 47.2 = 109.2°, so C ≈ 70.8°. All three add to 180° ✓

Tip!

Don't work out 2bc and cos A separately then subtract wrongly — put the whole line into the GDC at once. And never round before the final answer.

Area & Applications

Area of a triangle with sine, bearings & multi-triangle problems. ✎

① Area of a Triangle ✦ formula booklet

Area = ½ · a · b · sin C

the angle C sits BETWEEN sides a and b

② Area — Which Version?

2 sides + included angle

→ straight into ½ab sin C.

base + perpendicular height

→ the old ½ × base × height.

all 3 sides (SSS)

→ cosine rule for an angle first, then ½ab sin C.

③ Bearings

Measured clockwise from North, always written with three digits.

Write 055°, not 55°.

Back bearing (B to A) = bearing ± 180°.

Draw a North line at every point, then hunt for the triangle's interior angles.

④ Two-Triangle Problems

Find the shared side — it's the bridge between the two triangles.

Solve the triangle that has enough information first.

Carry the shared side into the second triangle.

Use angles on a straight line (180°) and angles at a point (360°) to fill gaps.

A triangular field has sides 12 m, 15 m and 20 m. Find its area.

SSS — no angle yet, so find one with the cosine rule. Let a = 20, b = 12, c = 15.

cos A = 12² + 15² − 20² / 2(12)(15) = −31 / 360

(obtuse — fine!)

Area = ½(12)(15) sin 94.9° ≈ 89.7 m²

negative cos

two sides that touch it