Many students find trigonometry scary at first. There are strange words, lots of numbers and triangles everywhere! However, once it is explained properly, it actually becomes quite logical and manageable.

At Apollo Scholars, we believe maths should feel clear, confidence-boosting and understandable. So this guide explains trigonometry slowly and simply, step-by-step.

In this article you will learn:

• Why sin 30° = 1/2 (in a really easy way)
• Useful facts about 30°, 45° and 60°
• What angle of depression means
• How to solve bearings problems
• Clear worked examples with answers

What Is SOH CAH TOA? A Simple GCSE Maths Guide (With Examples)

Why Does Sin 30° = 1/2?

Most students are told to simply “memorise” sin, cos and tan values, but that feels pointless if you do not understand them.

So let us explain why sin 30° = 1/2 in a way that actually makes sense.

🔺 Imagine an equilateral triangle

An equilateral triangle is one where:

• All sides are the same length
• All angles are 60°

Now imagine we cut it perfectly in half down the middle.

This gives us:

• A right-angled triangle
• With angles 30°, 60° and 90°

If the original triangle had sides of length 2:

• The hypotenuse becomes 2
• The small base becomes 1
• The tall side becomes √3 because:

1. Start with a right-angled triangle with:
   • Hypotenuse = 2
   • Short side = 1 (half the base of the equilateral triangle)
2. Use Pythagoras’ theorem:$${\text{hypotenuse}}^2={\text{base}}^2+{\text{height}}^2$$$$2^2=1^2+{\text{height}}^2$$
3. Simplify:$$4=1+{\text{height}}^2$$$${\text{height}}^2=3$$
4. Take the square root:$$\text{height}=\sqrt{3}$$

✅ So the tall side = √3

So in this simple triangle:$$\sin 30\mathrm{°}=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{1}{2}$$

That is it. No magic. No memorising. Just geometry.

Useful Trigonometry Values to Know

These values appear a lot in exams, so it helps to understand them.

Angle	sin	cos	tan
30°	1/2	√3 / 2	1 / √3
45°	√2 / 2	√2 / 2	1
60°	√3 / 2	1/2	√3

Even if the numbers look complicated, do not panic, we use calculators in exams!

Angle of Depression – Explained Simply

What does “angle of depression” mean?

It simply means:

• The angle when you look downwards from something high to something lower.

So imagine:

• You are standing on top of a flagpole
• You look down to a point on the ground
• That downward angle is the angle of depression

Question

A flagpole is 27 feet high.
The line of sight to a point on the ground is 34 feet.
Find the angle of depression to 1 decimal place.

This forms a right-angled triangle where:

• Opposite side = 27
• Hypotenuse = 34

We use sine because it links opposite and hypotenuse.

Step-by-Step

$$\sin \theta =\frac{27}{34}$$

$$\theta ={\sin }^{-1}\left(\frac{27}{34}\right)$$

$$\theta \approx 52.6\mathrm{°}$$

Final Answer

The angle of depression is 52.6°

Bearings – Explained Simply

Bearings are used in maps, ships, mountains, planes and navigation.

Bearings always:

• Start from North
• Go clockwise
• Use three numbers
  Example:
  045°, 180°, 270°, 291°

Remember:

• 270° = West
• Bearings bigger than 270° are “west but slightly towards north”

Bearings Problem 1

Dusty Mountain is 145 km west of Mount Frost
Its bearing is 287°
How far apart are the mountains?
Answer to the nearest km.

Final Answer

About 152 km apart

Bearings Problem 2

Silver Mountain is 165 km west of Mount Clether
Bearing = 291°
Find the distance between them
Nearest km.

Final Answer

About 177 km apart

Final Summary

In this guide you learned:

✔ Why sin 30° = 1/2 in a really easy way
✔ Useful trigonometry facts for 30°, 45° and 60°
✔ What angle of depression means and how to calculate it
✔ How bearings work and how to solve them
✔ Clear step-by-step examples with answers

If you or your child struggles with maths, trigonometry or exam confidence, Apollo Scholars offers friendly, expert tuition that makes learning clear, calm and stress-free.