Trigonometry for the TMUA

Trigonometry in the TMUA often requires applying relationships between sides and angles in triangles that are not right-angled. To do this, we use a standard labelling convention: corners are labelled with capital letters $A$, $B$, and $C$, while the sides opposite them are labelled with corresponding lower-case letters $a$, $b$, and $c$.

The Area of a Triangle

The most fundamental formula for the area of a triangle is $Area = \frac{1}{2} \times \text{base} \times \text{vertical height}$, or $\frac{1}{2}bh$.

This formula holds even if the top corner is not directly above the base. In such cases, one must be careful to use the vertical height above the horizontal base rather than a slanted height. This can be visualised by adding an identical triangle to form a parallelogram.

Using trigonometry, we can calculate the vertical height if it is not provided. In the diagram below, the height $h$ can be expressed as $b \sin C$. Substituting this into the area formula gives a version that uses two sides and the included angle:

$Area = \frac{1}{2}ab \sin C$

The Sine Rule

The sine rule is essentially a statement that the area of a triangle is constant regardless of which base and angle you choose for the calculation. Since the area can be written as $\frac{1}{2}ab \sin C$, $\frac{1}{2}bc \sin A$, or $\frac{1}{2}ca \sin B$, we can set them equal:

$\frac{1}{2}ab \sin C = \frac{1}{2}bc \sin A = \frac{1}{2}ca \sin B$

Multiplying by 2 and dividing by $abc$ yields the standard form of the sine rule:

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

Alternatively, it can be written as $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$. You use this rule when given two angles and one side (to find another side) or two sides and one non-included angle (to find another angle).

The Ambiguous Case of the Sine Rule

When using the sine rule to find an angle given two sides and a non-included angle, an 'ambiguous' result can occur. This happens because for any value of $\sin \theta$, there are two possible angles between $0^{\circ}$ and $180^{\circ}$ that satisfy it: $\theta$ and $180^{\circ} - \theta$.

Example: In triangle $ABC$, angle $A = 30^{\circ}$, side $b = 6$, and side $a = 4$. Find angle $B$.

Using the sine rule: $\frac{\sin B}{6} = \frac{\sin 30^{\circ}}{4}$ $\sin B = \frac{6 \sin 30^{\circ}}{4} = \frac{3}{4}$

Calculated values for $B$ are $48.6^{\circ}$ or $180^{\circ} - 48.6^{\circ} = 131.4^{\circ}$. Both are plausible because $30^{\circ} + 48.6^{\circ} < 180^{\circ}$ and $30^{\circ} + 131.4^{\circ} < 180^{\circ}$.

The Cosine Rule

The cosine rule acts as a more general form of Pythagoras' theorem. In a right-angled triangle, $a^2 = b^2 + c^2$.

If the angle $A$ is not $90^{\circ}$, we require a correction term. By considering the coordinates and projections of the sides, we derive:

$a^2 = b^2 + c^2 - 2bc \cos A$

This rule is used when you are given all three sides and want to find an angle, or when you are given two sides and the angle between them to find the third side. The rearranged form for finding an angle is $\cos A = \frac{b^2 + c^2 - a^2}{2bc}$.

Two and Three Dimensions

These rules are not limited to 2D triangles. In 3D problems, you should identify right-angled or non-right-angled triangles within the shape. Often, a side length calculated in one triangle (e.g., on the base of a pyramid) becomes a known side for a second triangle (e.g., a vertical cross-section).