Edexcel IGCSE Maths 复习笔记 4.9.2 3D Pythagoras & SOHCAHTOA

Edexcel IGCSE Maths 复习笔记 4.9.2 3D Pythagoras & SOHCAHTOA

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

 

6.5.1-3DPythagTrig-Notes-fig1-16.5.1-3DPythagTrig-Notes-fig1

 

  • 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

     

 

6.5.1-3DPythagTrig-Notes-fig2-16.5.1-3DPythagTrig-Notes-fig2

 

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

     

 

6.5.1-3DPythagTrig-Notes-fig3-16.5.1-3DPythagTrig-Notes-fig3-26.5.1-3DPythagTrig-Notes-fig3-3

 

  • 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

 

 

6.5.1-3DPythagTrig-Notes-fig4

 

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 planeThis should create a right-angled triangle

     

 

6.5.1-3DPythagTrig-Notes-fig5

6.5.1-3DPythagTrig-Notes-fig6

 

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

6.5.1-3DPythagTrig-Notes-fig7-16.5.1-3DPythagTrig-Notes-fig7-26.5.1-3DPythagTrig-Notes-fig7-3

 

Exam Tip

Add lines/triangles/etc. to any given diagram to help you see the problem and draw any 2D triangles separately as a 3D diagram can get hard to follow.

Worked Example

6.5.1-3DPythagTrig-Example-qu

Worked Example

6.5.1-3DPythagTrig-Example-sol-16.5.1-3DPythagTrig-Example-sol-2

转载自savemyexam

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