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Home Subjects General Mathematics Complex similarity & trig

Complex Similarity and Trig

General Mathematics

About these notes

Area Of Study

Geometry and measurement

Unit

Unit 4: General Mathematics Unit 4

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Complex Similarity and Trig

General Mathematics
01 May 2026

Similarity and Trigonometry in More Complex Contexts

Beyond Basic Right-Angled Triangles

Unit 4 extends geometry to situations where:
- Triangles are not right-angled (use Sine Rule or Cosine Rule)
- Problems require multiple steps combining similarity and trigonometry
- Three-dimensional figures require 2D cross-section analysis

The Sine Rule

For any triangle with sides \(a\), \(b\), \(c\) opposite angles \(A\), \(B\), \(C\):

\[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\]

Use when you know: two angles and one side (AAS or ASA), or two sides and an angle opposite one of them (SSA — beware the ambiguous case).

Worked Example

In triangle PQR: \(PQ = 12\) cm, \(\angle P = 55°\), \(\angle Q = 70°\). Find \(QR\).

\(\angle R = 180° - 55° - 70° = 55°\).

\[\frac{QR}{\sin P} = \frac{PQ}{\sin R} \implies QR = \frac{12 \sin 55°}{\sin 55°} = 12 \text{ cm}\]

(Triangle PQR is isosceles since \(\angle P = \angle R\).)

The Cosine Rule

For any triangle:

\[c^2 = a^2 + b^2 - 2ab\cos C\]
\[\cos C = \frac{a^2 + b^2 - c^2}{2ab}\]

Use when you know: three sides (SSS), or two sides and the included angle (SAS).

Worked Example

A triangular plot of land has sides 80 m and 65 m with an included angle of 42°. Find the third side.

\[c^2 = 80^2 + 65^2 - 2(80)(65)\cos(42°)\]
\[= 6400 + 4225 - 10400 \times 0.7431 = 10625 - 7728 = 2897\]
\[c = \sqrt{2897} \approx 53.8 \text{ m}\]

Multi-Step Similarity Problems

When scale factor \(k\) is applied, areas scale by \(k^2\) and volumes by \(k^3\).

Worked Example

Two similar cones have base radii 4 cm and 10 cm. The smaller cone has volume \(67.0 \text{ cm}^3\). Find the volume of the larger.

\[k = \frac{10}{4} = 2.5 \implies \text{Volume ratio} = 2.5^3 = 15.625\]
\[V_{\text{large}} = 67.0 \times 15.625 \approx 1047 \text{ cm}^3\]

3D Problems — Two-Stage Approach

Many 3D problems (e.g. angles of elevation in buildings, slope angles, diagonal measurements) require:
1. Identify a right-angled triangle in a 2D cross-section
2. Find a key length using Pythagoras or basic trig
3. Use that length in a second triangle (possibly non-right-angled)

COMMON MISTAKE: Applying basic SOH-CAH-TOA to a non-right-angled triangle. Always check for a right angle before using basic trig ratios. If no right angle, use Sine Rule or Cosine Rule.

EXAM TIP: For non-right-angled triangles, decide: known two angles + one side → Sine Rule; known two sides + included angle (or all three sides) → Cosine Rule.

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