3D Pythagoras & Trigonometry (AQA GCSE Further Maths)

3D Pythagoras & Trigonometry

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