Trigonometry in three dimensions

Introduction to Trigonometry in Three Dimensions

Welcome! If you have already mastered 2D trigonometry (like SOH CAH TOA, the Sine Rule, and the Cosine Rule), you are already 90% of the way there. 3D trigonometry isn't really about learning new formulas; it’s about learning how to "spot" flat, 2D triangles hiding inside 3D shapes like boxes (cuboids) and pyramids.

In this chapter, we will learn how to calculate lengths and angles within three-dimensional objects. Don't worry if you find it hard to "see" the shapes at first—we will use a step-by-step method to make it much simpler!

Important Note: According to your syllabus, you will not be asked to calculate the angle between two planes (flat surfaces) or the angle between a straight line and a plane. You only need to focus on lengths and angles within the triangles formed inside these shapes.

The Secret Weapon: 3D Pythagoras

In 2D, we use Pythagoras' Theorem (\(a^2 + b^2 = c^2\)) to find a missing side of a right-angled triangle. In 3D, we can use a "super-powered" version to find the long diagonal of a cuboid (a box).

If a cuboid has a length \(l\), a width \(w\), and a height \(h\), the length of the diagonal \(d\) from one corner to the opposite corner (through the middle of the box) is given by:

\(d^2 = l^2 + w^2 + h^2\)

Example: If a box is \(3\text{ cm}\) long, \(4\text{ cm}\) wide, and \(12\text{ cm}\) high, the diagonal through the middle is:
\(d^2 = 3^2 + 4^2 + 12^2\)
\(d^2 = 9 + 16 + 144 = 169\)
\(d = \sqrt{169} = 13\text{ cm}\)

Quick Review: The Tools in Your Toolbox

To solve 3D problems, you will pull out these 2D tools once you've found a triangle to work with:

• Right-angled triangles: Use \(\sin \theta = \frac{\text{opp}}{\text{hyp}}\), \(\cos \theta = \frac{\text{adj}}{\text{hyp}}\), \(\tan \theta = \frac{\text{opp}}{\text{adj}}\), or \(a^2 + b^2 = c^2\).
• Non-right-angled triangles: Use the Sine Rule \(\frac{a}{\sin A} = \frac{b}{\sin B}\) or the Cosine Rule \(a^2 = b^2 + c^2 - 2bc \cos A\).
• Area: The area of any triangle is \(\frac{1}{2}ab \sin C\).

Step-by-Step Strategy: "The Extraction"

The trick to 3D trig is to stop looking at the 3D shape and start looking for 2D "slices." Follow these steps:

1. Identify the goal: What length or angle are you looking for? Mark it clearly on the diagram.
2. Find the right triangle: Look for a triangle that includes your "goal" side/angle. Often, you will need to "create" this triangle by drawing a line through the shape.
3. Check for missing information: Often, you won't have enough info in your first triangle. You might need to solve a different triangle (usually on the base of the shape) first to find a shared side.
4. Redraw in 2D: This is the most important step! Draw the triangle flat on your paper, label the corners, and put in all the lengths and angles you know.
5. Calculate: Use your 2D trig rules to find the answer.

Did you know? Architects and video game designers use 3D trigonometry every day to make sure buildings stand up straight and characters move realistically in a 3D world!

Common 3D Shapes to Watch Out For

1. Cuboids (The Box)

In a cuboid, almost every corner is a right angle. If you need to find the diagonal across the floor, use the length and width with Pythagoras. If you need the diagonal through space, use the 3D Pythagoras formula mentioned above.

2. Pyramids

Square-based pyramids are very common in exams. The "height" (\(h\)) of the pyramid goes from the very top (the apex) straight down to the center of the base.

Top Tip: To find the height of a pyramid, you often have to find the distance from a corner of the base to the center of the base first. This usually involves doing Pythagoras on the base floor.

Common Mistakes to Avoid

• Assuming a triangle is right-angled: Just because it’s inside a box doesn't mean every triangle is a right-angled one. Always check where the lines meet!
• Rounding too early: If you find a length in Step 2, keep the full number in your calculator (or use the square root form) for Step 4. If you round too soon, your final answer will be slightly off.
• Misidentifying the "opposite" side: When you redraw your 2D triangle, be very careful which side is the hypotenuse (the longest side, across from the right angle).

Summary Table: Choosing Your Method

Key Takeaway: 3D Trigonometry is just 2D Trigonometry done twice. Find a triangle on the "floor" to get a shared side, then use that side to solve a "vertical" triangle. Keep your drawings clear, and always solve for the base first if you're stuck!