Chapter 9: Solution of Triangles

9.1 Sine Rule

The Sine Rule is used to solve non-right-angled triangles when given:

• Two angles and the length of one side (AAS or ASA).
• Two sides and a non-included angle (SSA).

Formula

For any triangle $ABC$ with side lengths $a$, $b$, and $c$ opposite to angles $A$, $B$, and $C$ respectively:

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

or equivalently, to find angles:

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

The Ambiguous Case

When given two sides ($a$ and $b$) and a non-included acute angle ($A$ opposite to side $a$), an ambiguous case occurs if $a < b$ and $a > b \sin A$ (or $h < a < b$ where $h = b \sin A$ is the height of the triangle):

• Two distinct triangles can be formed with the given measurements.
• The first angle $B_1$ is acute: $B_1 = \sin^{-1}\left(\frac{b \sin A}{a}\right)$.
• The second angle $B_2$ is obtuse: $B_2 = 180^\circ - B_1$.

9.2 Cosine Rule

The Cosine Rule is used to solve triangles when given:

• Two sides and the included angle (SAS).
• All three side lengths (SSS).

Formula for Side Lengths

$$a^2 = b^2 + c^2 - 2bc \cos A$$ $$b^2 = a^2 + c^2 - 2ac \cos B$$ $$c^2 = a^2 + b^2 - 2ab \cos C$$

Formula for Interior Angles

$$\cos A = \frac{b^2 + c^2 - a^2}{2bc}$$ $$\cos B = \frac{a^2 + c^2 - b^2}{2ac}$$ $$\cos C = \frac{a^2 + b^2 - c^2}{2ab}$$

9.3 Area of a Triangle

Included Angle Formula

The area of triangle $ABC$ when two sides and their included angle are known:

$$\text{Area} = \frac{1}{2} ab \sin C = \frac{1}{2} ac \sin B = \frac{1}{2} bc \sin A$$

Heron's Formula

When all three sides $a$, $b$, and $c$ are known, the area can be found without calculating angles using Heron's Formula:

$$\text{Area} = \sqrt{s(s - a)(s - b)(s - c)}$$

where $s$ is the semi-perimeter of the triangle:

$$s = \frac{a + b + c}{2}$$

9.4 Application of Sine Rule, Cosine Rule, and Area Formula

Real-world applications involve solving 2D and 3D geometric problems, bearings, elevations, and practical measurement tasks by strategically choosing between the Sine Rule, Cosine Rule, and Area Formulas.