3D Pythagoras & Trigonometry | Mathematics for GCSE/IGCSE - Year 11 PDF Download

3D Pythagoras & Trigonometry

• 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 = Opposite ÷ Hypotenuse
  • Cosine – CAH – cos θ = Adjacent ÷ Hypotenuse
  • Tangent – TOA – tan θ = Opposite ÷ Adjacent

How does Pythagoras work in 3D?

• 3D shapes can often be deconstructed into multiple 2D shapes, such as nets and surface areas.
• When applying Pythagoras' Theorem in 3D, focus on identifying right-angled triangles with two known sides and one unknown side.

• The 3D version of the Pythagorean Theorem is expressed as d² = x² + y² + z².
• It is often simpler to solve 3D problems by breaking them into multiple 2D problems for better visualization.

How does SOHCAHTOA work in 3D?

• Once more, seek out a combination of right-angled triangles that could help in determining the missing angle or side.
• In three-dimensional scenarios, the angle being dealt with might pose challenges.
  • Determining the angle between a line and a plane is not always straightforward.
  • If uncertainty arises, place a point on the line and extend a new line to the plane.
  • This approach should establish a right-angled triangle, aiding in the solution process.

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