- 1. The Sine Rule — Finding Missing Angles
Rearrange the sine rule as $\frac{\sin A}{a} = \frac{\sin B}{b}$ to make an angle the subject, recognise the SSA pattern, find a missing angle with inverse sine, and understand that $\sin^{-1}$ returns only the acute angle.
- 2. Two Triangles or One — The Ambiguous Case, and Area = ½ab sin C
Use the swinging-stick idea to decide whether an SSA problem gives zero, one, or two triangles, find and test both solutions in an ambiguous case, derive $\text{Area} = \frac{1}{2}ab\sin C$ from the altitude $h = b\sin C$, and use it to find an area or an enclosed angle.
- 3. The Cosine Rule — Finding a Missing Side
Explain why the sine rule fails for SAS, follow the derivation of the cosine rule for acute and obtuse triangles, state all three forms, and apply $a^2 = b^2 + c^2 - 2bc\cos A$ to find the third side when two sides and the included angle are known.
- 4. The Cosine Rule for Angles — and the "Which Rule?" Strategy
Rearrange the cosine rule to $\cos A = \frac{b^2 + c^2 - a^2}{2bc}$ to find an angle from three sides (SSS), read a negative cosine as an obtuse angle, and apply a decision framework to choose between the sine and cosine rules.
- 5. Bearings — North Lines and the Hidden Triangle
Read and write three-figure bearings, compute back-bearings, draw parallel north lines at every point, and use alternate and co-interior angles to extract the interior angle of the hidden triangle at each vertex.
- 6. Bearings and Distances — Applying the Sine Rule
Build a bearings diagram, recognise when it gives an AAS/ASA pattern, apply the sine rule to find an unknown distance, and convert an interior triangle angle back into a final three-figure bearing.
- 7. Bearings and Distances — Applying the Cosine Rule, and Reverse Bearings
Recognise SAS and SSS bearings problems and apply the cosine rule to find a missing distance from two distances and an included angle, combine the cosine and sine rules to find a bearing, and handle the interior-versus-reflex angle decision when directions at a vertex are far apart.
- 8. Angles of Elevation and Depression — When the Triangle Is Not Right-Angled
Recall the angle of elevation and depression and their equality as alternate angles, recognise when two observation points create a non-right triangle, find a height with the two-observer method (sine rule then a right-triangle step), and choose the right rule in mixed elevation, depression and height problems.
