Mathematics — Geometry: Trigonometry (for 14‑year‑olds, Kenya)

This page gives easy notes on basic trigonometry you will meet in school. We use simple language, diagrams and examples you can work through with a scientific calculator set in degrees.

1. What is trigonometry?

Trigonometry studies relationships between the angles and sides of triangles. In school, we start with right-angled triangles (one angle is 90°).

2. Key words

• Hypotenuse — the longest side, opposite the 90° angle.
• Opposite — the side opposite the angle we are considering (θ).
• Adjacent — the side next to the angle (θ), not the hypotenuse.

3. SOHCAHTOA (easy memory tool)

For an angle θ in a right triangle:

• sin θ = Opposite / Hypotenuse
• cos θ = Adjacent / Hypotenuse
• tan θ = Opposite / Adjacent

Remember: SOH-CAH-TOA.

4. Diagram (right triangle)

5. How to use SOHCAHTOA — Examples

Example 1 — Find a side

Given right triangle with angle θ = 30° and hypotenuse = 10 cm. Find the opposite side (BC).

Use sin: sin θ = Opposite / Hypotenuse ⇒ Opposite = Hypotenuse × sin θ

Opposite = 10 × sin 30° = 10 × 0.5 = 5 cm

Answer: 5 cm

Example 2 — Find an angle

In a right triangle, opposite = 7 cm and adjacent = 24 cm. Find angle θ.

Use tan: tan θ = Opposite / Adjacent = 7 / 24 ≈ 0.2917

θ = arctan(0.2917). On the calculator (DEGREE mode): θ ≈ 16.26°

Answer: θ ≈ 16.3°

6. Special angles (values you should learn)

Angle	sin	cos	tan
30°	1/2	√3/2 ≈ 0.866	1/√3 ≈ 0.577
45°	√2/2 ≈ 0.707	√2/2 ≈ 0.707	1
60°	√3/2 ≈ 0.866	1/2	√3 ≈ 1.732

7. Important identity

For any angle θ in a right triangle: sin²θ + cos²θ = 1

This helps when you know one of sin or cos and want the other. Example: if sin θ = 3/5, then cos θ = √(1 − (3/5)²) = 4/5.

8. Using a calculator — tips

• Always set the calculator to DEGREE mode for school problems unless told otherwise.
• To find an angle: use inverse functions: sin⁻¹, cos⁻¹, tan⁻¹ (often labelled asin, acos, atan).
• Round answers to one or two decimal places as required by the question.

9. Word problem example (Kenyan context)

A ladder is placed against a wall so that the top reaches a height of 4 m. The ladder makes an angle of 60° with the ground. How long is the ladder?

Let ladder = hypotenuse. Use sin 60° = opposite / hypotenuse ⇒ hypotenuse = opposite / sin 60°

Length = 4 / sin 60° = 4 / (√3/2) = 4 × 2/√3 = 8/√3 ≈ 4.62 m (to 2 d.p.)

Answer: ≈ 4.62 m

10. Practice questions

1. In a right triangle, hypotenuse = 13 cm, adjacent = 5 cm to angle θ. Find θ.
2. Find the opposite side if θ = 45° and adjacent = 7 cm.
3. A tree casts a shadow 10 m long. The angle of elevation of the sun is 30°. How tall is the tree?

Answers

1. cos θ = adjacent/hypotenuse = 5/13 ⇒ θ = cos⁻¹(5/13) ≈ 67.38°
2. tan 45° = opposite/adjacent = 1 ⇒ opposite = adjacent = 7 cm
3. tan 30° = height / 10 ⇒ height = 10 × tan 30° = 10 × 0.577 ≈ 5.77 m

11. Common mistakes — avoid them!

• Forgetting to use degree mode on the calculator. (Results then are wrong!)
• Mixing up opposite and adjacent — always mark the angle first and label sides.
• Using sin when the question needs tan (or vice versa). Check which sides are given.

12. Quick study checklist

• Learn SOHCAHTOA and practice labeling triangles.
• Memorise special values for 30°, 45°, 60°.
• Practice using inverse trig functions to find angles.
• Work through real problems (ladders, heights and distances).

Good luck! Practice a few questions every week — that will make trigonometry easy and useful for geometry problems in KCSE and everyday life.