U26 Applications of Trigonometry: Essential Formulas

True Bearing

The true bearing of a point $B$ from a point $A$ is the angle measured in degrees, from north at $A$, in a clockwise direction to the line joining $A$ to $B$. It is always expressed as a three-digit number between $000^\circ$ and $360^\circ$.

Key Definitions

The angle of elevation is the angle between the horizontal line of sight and the line of sight to an object above the observer. The angle of depression is the angle between the horizontal line of sight and the line of sight to an object below the observer. They are alternate angles when lines are parallel.

Lengths and Angles in 3D Shapes

To find lengths or angles in three-dimensional figures (e.g., pyramids, wedges), you often need to identify right-angled triangles within the 3D structure. Apply Pythagoras' Theorem $a^2 + b^2 = c^2$ and trigonometric ratios $\sin \theta$, $\cos \theta$, $\tan \theta$ in these triangles.

Angle Between a Line and a Plane

The angle $\theta$ between a line and a plane is defined as the angle between the line and its orthogonal projection onto that plane. It is the complement of the angle between the line and the normal to the plane.

Choosing the Correct Rule

For any triangle $ABC$ with sides $a$, $b$, $c$ opposite angles $A$, $B$, $C$ respectively:

Use the Sine Rule when you know either: (i) Two angles and one side, or (ii) Two sides and a non-included angle (the ambiguous case).

Use the Cosine Rule when you know either: (i) Three sides, or (ii) Two sides and the included angle.

Formula Using Sine

The area of triangle $ABC$ can be found if you know two sides and the included angle.