Why heights and distances matter
Ever wondered how to find the height of a tree without climbing it? Or how a surveyor measures a building from the ground? Trigonometry lets us do exactly that, using angles and a little bit of geometry.
💡 In Simple Words: When you know an angle and one side of a right‑angled triangle, you can figure out the other sides. That’s the core trick behind every heights‑and‑distances problem.
Understanding the basics
First, picture a right triangle – a shape with one 90° corner, just like the corner of a book. The longest side opposite the right angle is called the hypotenuse. The two shorter sides are the adjacent side (next to the angle you’re looking at) and the opposite side (the one across from that angle).
In a heights‑and‑distances question you’ll often see an angle of elevation (the angle looking up) or an angle of depression (the angle looking down). Think of it like looking up at a kite or down at a car from a balcony.
Key trigonometric ratios
Trigonometry gives us three handy ratios that connect angles to side lengths:
- sin θ = opposite ÷ hypotenuse
- cos θ = adjacent ÷ hypotenuse
- tan θ = opposite ÷ adjacent
Remember the old “SOH‑CAH‑TOA” rhyme? It’s just a fun way to remember these three formulas.
Step‑by‑step method for word problems
Solving a heights‑and‑distances problem is like following a recipe. Here’s a simple flow you can copy every time:
Let’s break each step down.
1. Read the question carefully
Spot the unknown – is it a height, a distance, or an angle? Highlight the numbers given.
2. Draw a clear diagram
A sketch turns words into a picture. Draw the right triangle, label the angle of elevation or depression, and write the known lengths.
3. Mark known angles & lengths
Put the numbers you have on the diagram. If the problem says “the angle of elevation is 30° and the distance from the point to the base is 50 m”, write 30° at the appropriate corner and 50 m on the adjacent side.
4. Choose the right trig ratio
Ask yourself: Which side are you looking for? If you need the height (the opposite side) and you know the adjacent side, use tan θ. If you have the hypotenuse, use sin θ or cos θ as needed.
5. Do the calculation
Plug the numbers into the formula and solve. Don’t forget to keep your calculator in degree mode for angles measured in degrees.
6. Check the answer
Does the result make sense? A height can’t be negative, and a distance should be realistic compared to the given numbers.
Worked example 1: Finding a tree’s height
Problem: From a point 40 m away, the angle of elevation to the top of a tree is 35°. What is the height of the tree?
Solution:
- Draw a right triangle. The ground distance (40 m) is the adjacent side, the tree height is the opposite side, and the angle at the ground point is 35°.
- We need the opposite side, so we use tan θ = opposite ÷ adjacent.
- tan 35° ≈ 0.7002. Set up the equation: 0.7002 = height ÷ 40.
- Multiply both sides by 40: height = 0.7002 × 40 ≈ 28.0 m.
So the tree is about 28 m tall.
Worked example 2: Distance between two towers
Problem: Two towers stand on a level ground. From the top of the left tower, the angle of depression to the base of the right tower is 20°. The height of the left tower is 60 m. Find the horizontal distance between the towers.
Solution:
- Draw a right triangle with the left tower’s top as the observation point. The angle of depression (20°) is measured downwards, which means the angle of elevation from the base of the right tower to the observation point is also 20°.
- The opposite side of this triangle is the height of the left tower (60 m). The adjacent side is the distance we want.
- Use tan θ = opposite ÷ adjacent: tan 20° ≈ 0.36397 = 60 ÷ distance.
- Rearrange: distance = 60 ÷ 0.36397 ≈ 165 m.
The towers are roughly 165 m apart.
Quick reference table
| What you know | What you need | Use this ratio |
|---|---|---|
| Angle + adjacent side | Opposite side (height) | tan θ = opposite ÷ adjacent |
| Angle + opposite side | Adjacent side (distance) | tan θ = opposite ÷ adjacent → adjacent = opposite ÷ tan θ |
| Angle + hypotenuse | Opposite side | sin θ = opposite ÷ hypotenuse → opposite = hypotenuse × sin θ |
| Angle + hypotenuse | Adjacent side | cos θ = adjacent ÷ hypotenuse → adjacent = hypotenuse × cos θ |
Common pitfalls and how to avoid them
- Mixing up opposite and adjacent: Always label your diagram before plugging numbers.
- Using the wrong unit: Keep everything in the same unit (metres, feet, etc.) before solving.
- Calculator in radian mode: Most school problems use degrees, so double‑check the mode.
📝 Likely Exam Questions
- Question: From a point 30 m away from the base of a flagpole, the angle of elevation to its top is 45°. Find the height of the flagpole.
Answer: Use tan 45° = 1 = height ÷ 30 → height = 30 m. - Question: The angle of depression from the top of a 50 m high lighthouse to a boat is 10°. Calculate the horizontal distance of the boat from the lighthouse base.
Answer: tan 10° = 50 ÷ distance → distance = 50 ÷ tan 10° ≈ 284 m. - Question: Two poles of equal height stand 80 m apart. From the top of one pole, the angle of depression to the base of the other is 30°. Find the height of each pole.
Answer: tan 30° = height ÷ 80 → height = 80 × tan 30° ≈ 46.2 m.