3D Pythagoras & Trigonometry (CIE IGCSE Maths: Extended)

What are Pythagoras & SOHCAHTOA?

• Pythagoras’ Theorem helps us find missing side lengths of a right-angled triangle
• It is also frequently used for finding the distance (or length) of a line
• SOHCAHTOA is an acronym for the three trigonometric ratios that connect angles (θ) and sides (Opposite, Hypotenuse and Adjacent) in a right-angled triangle
  • Sine – SOH – sin θ = O ÷ H
  • Cosine – CAH – cos θ = A ÷ H
  • Tangent – TOA – tan θ = O ÷ A

How does Pythagoras work in 3D?

• 3D shapes can often be broken down into several 2D shapes
  • For example nets and surface area

• With Pythagoras’ Theorem problems you will be specifically looking for right‑angled triangles
  • The right-angled triangles you need will have two known sides and one unknown side

• There is a 3D version of the Pythagoras’ Theorem formula
  • d² = x² + y² + z²

• However it is usually far easier to see a problem by splitting it into two or more 2D problems

How does SOHCAHTOA work in 3D?

• Again look for a combination of right-angled triangles that would lead to the missing angle or side
• The angle you are working with can be awkward in 3D
  • The angle between a line and a plane is not obvious
  • If unsure, put a point on the line and draw a new line to the plane

    This should create a right-angled triangle

• Once you have your 2D triangle(s) you can begin to solve problems