3D Trigonometry Notes

3D Trigonometry Overview

  • Focus on applying trigonometry in three-dimensional problems.
  • Essential to visualize and separate parts of the problem in two dimensions.
  • Quick sketches are helpful during problem-solving.

Key Concepts

1. Cuboid Geometry

  • Consider a cuboid ABCDEFGH with dimensions given.
  • Useful dimensions:
    • Length: $AG$
    • Sides: AB = 6 cm, AD = 5 cm, etc.
  • Calculating lengths and angles:
    • Length can be calculated using Pythagorean theorem in 2D.
    • Example: Finding length $BD$ using base dimensions.
    • Formula: $BD = ext{sqrt}(7^2 + 6^2)$ gives $BD = ext{sqrt}(85) \approx 9.22$ cm.

2. Finding Angles

  • To find angles, visualize angles within a right triangle.
  • Use the tangent function to determine angles:
    • Example: For angle $ heta$ between line $BH$ and the base plane:
    • Use: an( heta) = \frac{ ext{Opposite}}{ ext{Adjacent}} ; heta = \tan^{-1}\left(\frac{6}{BD}\right)
    • Substitute: heta = \tan^{-1}\left(\frac{6}{\sqrt{65}}\right) \approx 36.7°

3. Solving Practical Problems: The Great Pyramid of Giza

  • Pyramid dimensions: base = 230 m, height = 139 m.
    • Walking Distance from Corner to Top:
    • Calculate diagonal length of the base facing (half of base length involves Pythagorean theorem). The diagonal is:
      d = \text{sqrt}(230^2 + 230^2) = \text{sqrt}(105800) \approx 325.27 \text{ m}
    • Create a right triangle using the height and half the diagonal:
    • Height (Opposite): 139 m
    • Half diagonal (Adjacent): 162.63 m
    • Angle $ heta$ calculation:
      \tan\theta = \frac{139}{162.63} \Rightarrow \theta \approx 40.5°
  • The total distance walked is:
    • Using Pythagorean theorem for the entire path:
      d_{total} = \text{sqrt}(139^2 + 162.63^2) \approx 213.9 ext{ m}

4. Important Techniques

  • Diagrams are crucial in visualizing 3D shapes and extracting 2D problems.
  • Label dimensions clearly and ensure accuracy in calculations.
  • Rounding: Be careful especially in crucial steps, like determining angles or lengths.

Summary of Learning Points

  • Utilize 2D sketches to resolve 3D trigonometry problems effectively.
  • Familiarize with the use of trigonometric functions to find angles and lengths in various configurations. 2D representations simplify complex 3D relationships into manageable forms.