Trigonometry Made Simple: Sin, Cos, and Tan Explained

What Is Trigonometry?

Trigonometry is the branch of mathematics that deals with the relationships between the sides and angles of triangles. It's one of the most useful areas of maths — used everywhere from engineering to music to game development.

In South African schools, you'll start encountering trigonometry from Grade 10 onwards, and it becomes a major part of Grade 11 and 12 maths.

The Three Basic Ratios

In a right-angled triangle, the three trigonometric ratios are defined relative to an angle θ (theta):

• sin θ = opposite / hypotenuse
• cos θ = adjacent / hypotenuse
• tan θ = opposite / adjacent

The Memory Trick: SOH-CAH-TOA

• Sin = Opposite / Hypotenuse
• Cos = Adjacent / Hypotenuse
• Tan = Opposite / Adjacent

Say it out loud a few times: "SOH-CAH-TOA". You'll never forget it.

Identifying the Sides

Before using any ratio, you need to correctly identify which side is which:

1. Hypotenuse — always the longest side, opposite the right angle
2. Opposite — the side across from the angle you're working with
3. Adjacent — the side next to the angle you're working with (that isn't the hypotenuse)

The opposite and adjacent sides change depending on which angle you're looking at!

Worked Example

In a right triangle where the angle is 30°, the opposite side is 5 cm. Find the hypotenuse.

Solution:

• We know the opposite and want the hypotenuse → use sin
• sin 30° = 5 / hypotenuse
• 0.5 = 5 / hypotenuse
• hypotenuse = 5 / 0.5 = 10 cm

Special Angles

These angles come up constantly. Memorise them:

| Angle | sin | cos | tan | |-------|-----|-----|-----| | 0° | 0 | 1 | 0 | | 30° | 1/2 | √3/2 | 1/√3 | | 45° | √2/2 | √2/2 | 1 | | 60° | √3/2 | 1/2 | √3 | | 90° | 1 | 0 | undefined |

Common Mistakes

1. Mixing up opposite and adjacent — always label relative to YOUR angle
2. Calculator in wrong mode — make sure it's set to degrees, not radians
3. Forgetting to simplify — always give exact answers where possible
4. Not drawing a diagram — a quick sketch saves time and prevents errors

What's Next?

Once you've mastered the basics, you'll move on to trigonometric identities, equations, and the sine and cosine rules for non-right triangles. These build directly on what you've learned here.

Struggling With Trig?

Book a session and we'll work through it from the basics until you're confident. No judgment — everyone finds trig tricky at first.