🧠 Chapter 7 (Extended): Pythagoras’ Theorem and Trigonometry in 3D

Welcome to the exciting world of three-dimensional geometry! Don't worry if shapes like cuboids and pyramids look scary at first. This chapter is all about taking those complicated 3D shapes and cleverly breaking them down into simple, familiar 2D right-angled triangles. Once you master the trick of finding the right triangle, the rest is just using Pythagoras' theorem and SOH CAH TOA, just like you did in 2D problems!

1. Your Essential 2D Toolkit (Review)

Before jumping into 3D, let’s quickly refresh the two main tools you will be using repeatedly. Remember, every 3D problem must be solved by finding a right-angled triangle inside the figure.

a. Pythagoras' Theorem (Finding Lengths)

Used to find the length of a side in a right-angled triangle when you know the other two sides.

The formula is: \(\mathbf{a^2 + b^2 = c^2}\), where \(c\) is always the longest side (the hypotenuse).

b. Trigonometry Ratios (SOH CAH TOA)

Used to find unknown sides or angles when you have a right-angled triangle and at least one angle (other than 90°) and one side.

  • SOH: \(\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}\)
  • CAH: \(\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}\)
  • TOA: \(\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}\)

🧠 Memory Aid: SOH CAH TOA (or a phrase like *S*ome *O*ld *H*en *C*aught *A*nother *H*en *T*aking *O*ff *A*lways).

Key Takeaway: 3D geometry is just 2D trigonometry and Pythagoras applied multiple times. Your first step is always to find or draw the correct right-angled triangle.

2. Applying Pythagoras in 3D (Finding Diagonals)

In 3D shapes like cuboids, we often need to find the distance between two corners that are not on the same face. This involves using Pythagoras' theorem twice.

Step-by-Step: Finding the Space Diagonal of a Cuboid

Imagine a cuboid with length (L), width (W), and height (H). We want to find the space diagonal (\(D\)) running from the bottom front corner to the top back corner.

Step 1: Find the Diagonal on the Base (The Floor)

First, look only at the bottom face (the rectangle defined by L and W). This diagonal (\(d\)) is the hypotenuse of a right-angled triangle on the floor.

$$\text{Diagonal on Base } (d^2) = L^2 + W^2$$

Step 2: Find the Space Diagonal (The Main Diagonal)

Now, consider a second right-angled triangle. Its sides are:

  • Side 1: The height of the cuboid (\(H\)).
  • Side 2: The diagonal of the base (\(d\)) you just calculated.
  • Hypotenuse: The space diagonal (\(D\)).

$$\text{Space Diagonal } (D^2) = d^2 + H^2$$

Substituting Step 1 into Step 2 gives the general 3D Pythagoras formula:

$$\mathbf{D^2 = L^2 + W^2 + H^2}$$

Example Analogy: Think of a spider walking from one corner of a room (floor level) to the opposite corner of the ceiling. It has to cross the floor (Step 1) and then climb the height of the room (Step 2).

⚠️ Common Mistake Alert!

Do NOT round your intermediate answer (\(d\)) in Step 1. Use the full value (or \(d^2\)) directly in Step 2 to ensure your final answer (\(D\)) is accurate to 3 significant figures.

Key Takeaway: To find a 3D length, you often need to solve two consecutive 2D right-angled triangles.

3. Trigonometry in 3D: Identifying the Angle

Finding an angle in a 3D shape requires you to identify the correct right-angled triangle that contains that angle. Often, one of the required sides of this triangle must first be calculated using Pythagoras (see Section 2).

Step-by-Step: Solving a 3D Trigonometry Problem

Let’s say you need to find the angle \(\theta\) in a cuboid.

Step 1: Draw the 3D Shape and Mark the Angle

Clearly label all known lengths (L, W, H) and the angle \(\theta\) you need to find.

Step 2: Identify the Right-Angled Triangle that Holds \(\theta\)

This is the crucial step. Look for the 90° corner. If the angle is formed by a line lying on a face and a line coming out of the face (like a height), the right angle will be on the face itself.

Step 3: Calculate Missing Side Lengths (using Pythagoras)

Does your identified triangle have enough side lengths (at least two) to use SOH CAH TOA? If not, use Pythagoras on another face (the floor or a wall) to find a missing side first.

Step 4: Apply SOH CAH TOA

Once you have two sides, determine which ratio (sin, cos, or tan) relates the angle \(\theta\) to the known sides. Then use the inverse function (\(\sin^{-1}\), \(\cos^{-1}\), or \(\tan^{-1}\)) to find \(\theta\).

Remember: Angles should usually be given correct to 1 decimal place unless otherwise specified.

4. The Angle Between a Line and a Plane (E7.6 Specific)

This is often the most important type of 3D trigonometry problem. You must know how to correctly define and calculate this angle.

a. Definition: What is the Angle?

The angle between a line and a plane is the angle between the line and its projection onto the plane.

Projection Analogy (The Shadow Trick)

Imagine the plane is the flat ground (like the floor of a room) and the line is a pole sticking down from the ceiling (like a ramp). If you shine a light directly from above the pole, the projection is the shadow the pole casts on the floor.

  • The right angle is always between the line's shadow (the projection) and the vertical height.
  • The angle you need is always located where the line meets its projection.
b. Identifying the Key Components

If you have a line AB and a plane P:

  1. The line AB meets the plane P at point B. (This is where the angle is located.)
  2. The point A is lifted above the plane P.
  3. The projection is the shadow of the line AB onto the plane. To find it, drop a perpendicular line from A straight down to the plane, meeting it at point C.

The required angle is \(\angle\text{ABC}\).

The right-angled triangle is \(\triangle\text{ABC}\), with the right angle at C.

  • The Opposite side is the height AC.
  • The Adjacent side is the projection BC.
  • The Hypotenuse is the line AB itself.
Example: Angle of Elevation

If a question asks for the angle of elevation of a point (A) from another point (B), this is exactly the angle between the line AB and the horizontal plane (the ground/floor). The angle you are looking for is the one inside the ground, pointing up to the line.

Quick Review Box: Angle Between Line and Plane

To find the angle \(\theta\):

  1. Identify the line.
  2. Identify the plane (the floor).
  3. Find the "shadow" of the line on the floor.
  4. The angle is in the right-angled triangle formed by the Line, the Height, and the Shadow.

Key Takeaway: The angle between a line and a plane is the angle between the line and the shadow it casts. Always construct a right-angled triangle using the vertical height and the horizontal projection.