U42 Applications of Trigonometry (3D) - Essential Formulas

Angle Between a Line and a Plane

The angle $\theta$ between a line and a plane is defined as the acute angle between the line and its orthogonal projection onto the plane. If $\alpha$ is the angle between the line and the normal to the plane, then $\theta = 90^\circ - \alpha$.

Angle Between Two Planes

The angle $\phi$ between two intersecting planes is defined as the acute angle between their normal vectors $ \mathbf{n_1} $ and $ \mathbf{n_2} $. It can be found using the dot product formula.

Strategy for Solving 3D Problems

Identify relevant 2D triangles within the 3D figure (often right-angled triangles). Apply the sine rule $ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} $ or the cosine rule $ a^2 = b^2 + c^2 - 2bc\cos A $ to these triangles to find unknown lengths or angles.

In a typical 3D pyramid problem, you might first find a side length in the base triangle using the cosine rule, then use that length in a vertical right-angled triangle to find the height.

Key Definitions

True Bearing: The clockwise angle from north, measured in degrees (e.g., $ 075^\circ T $).

Angle of Elevation: The angle from the horizontal upward to an object.

Angle of Depression: The angle from the horizontal downward to an object. Note: The angle of elevation from point $A$ to point $B$ equals the angle of depression from $B$ to $A$.