3D Trigonometry

Study Notes

3D trigonometry involves applying trigonometric skills to three-dimensional shapes to find missing sides or angles. It requires understanding which rules and formulae to use for different angles and side lengths.

• Sine, Cosine, and Tangent Ratios — used to relate angles and sides in right-angled triangles. Example: In a right-angled triangle, $\sin(\theta) = \frac{opposite}{hypotenuse}$.
• Pythagoras’ Theorem — relates the lengths of the sides in a right-angled triangle. Example: In a triangle with sides $a$, $b$, and hypotenuse $c$, $a^2 + b^2 = c^2$.
• Angle Between a Line and a Plane — the angle formed between a line and a plane in 3D space. Example: Calculating the angle between a diagonal of a cuboid and its base.

Exam Tips

Key Definitions to Remember

• Sine, cosine, and tangent ratios
• Pythagoras’ theorem
• Angle between a line and a plane

Common Confusions

• Mixing up which trigonometric ratio to use
• Forgetting to use 3D projections to simplify problems

Typical Exam Questions

• Calculate the length of a diagonal in a cuboid? Use Pythagoras’ theorem in 3D.
• Find the angle between a line and the base of a cuboid? Use trigonometric ratios with a 2D projection.
• Determine the length of a side in a triangular prism? Apply Pythagoras’ theorem and trigonometric ratios.

What Examiners Usually Test

• Ability to apply trigonometric ratios in 3D
• Correct use of Pythagoras’ theorem in three dimensions
• Understanding of angles between lines and planes