3D Pythagoras & Trigonometry (AQA GCSE Further Maths)

What skills will I need to solve 3D problems involving Pythagoras & Trigonometry?

• Pythagoras’ Theorem helps us find missing side lengths of a right-angled triangle; 
  • It is also frequently used for finding the distance/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 –
  • cosine – CAH – 
  • tangent – TOA – 
• The hardest problems may involve sine rule, cosine rule and/or area of a triangle
  • sine rule - 
  • cosine rule - 
  • area of a triangle - 
• Other skills involving angles may also be involved
  • e.g. bearings, angles of elevation and depression, angles in parallel and perpendicular lines

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

How does trigonometry work in 3D?

• Again look for a combination of triangles that would lead to the missing angle or side
  • Ideally look for right-angled triangles so SOHCAHTOA can be used
  • Failing that, look for 'pairs' to use sine rule and 'two sides and the angle between them' for cosine rule
• 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