## Three-dimensional Trigonometry

Trigonometry can be applied to solid, three-dimensional shapes such as cuboids, pyramids and triangular prisms.

### Definitions

• plane is a flat surface.
• Two lines or planes are perpendicular if they meet at right angles.
• Collinear points lie on the same line.
• Coplanar points lie on the same plane.
• Concurrent lines pass through the same point.
• polyhedron is a solid shape having several faces. e.g. a pyramid.
• prism is a polyhedron with a regular cross-section. e.g. a cuboid.
• An edge is the intersection of two faces.
• vertex is the point of intersection of three or more faces of a polyhedron.

### Intersecting Planes

 Two planes intersect at a line. To find the angle between two planes, draw lines on each plane that meet on, and are perpendicular to, the line of intersection. e.g. Angle ABC is the angle between the two planes. ### Line and plane

To find the angle between a line and a plane, a perpendicular line is dropped from any point on the line. The point where this perpendicular meets the plane is then joined to the point where the line meets the plane. This line is called the projection of the line on the plane.

The angle required is the angle between the projection and the line.

e.g. CB is the projection of line AB on the plane. The angle between line AB and plane p is ABC

### Problems

When a three-dimensional problem has to be solved, the best way is to try and isolate and redraw the triangles involved.

e.g. To find the length of FD

 Redraw the triangle FDGNow use Pythagoras' Theorem:  To find angle CFD

Redraw the triangle CFD
Now use SOH/CAH/TOA (to 1 d.p.) 