Trigonometric Ratios
Engineering Services (ESE) – Mathematics – Beginner Notes
1. What are Trigonometric Ratios?
Think of a right‑angled triangle (a triangle with one 90° corner).
The trigonometric ratios tell you how the three sides relate to each other.
| Name | Everyday Analogy | What it Means |
|---|---|---|
| Sine | “Up‑and‑down” of a swing | Opposite side ÷ Hypotenuse |
| Cosine | “Side‑by‑side” of a ladder | Adjacent side ÷ Hypotenuse |
| Tangent | “How steep it is” | Opposite side ÷ Adjacent side |
Later we’ll add the other three (cosecant, secant, cotangent) – they are just the reciprocals (inverse) of the first three.
2. The Right‑Angle Triangle
/|
/ | a ← adjacent
/ |
/___|
b c ← hypotenuse (the longest side)
- Opposite side – the side that is opposite the angle you’re looking at.
- Adjacent side – the side that touches the angle but isn’t the hypotenuse.
- Hypotenuse – the side opposite the right angle (always the longest).
3. The Three Main Ratios
| Ratio | Formula (for angle θ) | Everyday Picture |
|---|---|---|
| sin θ | Opposite ÷ Hypotenuse | A swing’s height compared to its length. |
| cos θ | Adjacent ÷ Hypotenuse | The horizontal distance along the ground compared to the ladder length. |
| tan θ | Opposite ÷ Adjacent | How steep a ramp is (rise over run). |
Tiny Example
You have a ladder that leans against a wall.
The ladder (hypotenuse) is 5 m long.
The wall is 3 m high (opposite side).
Find the angle the ladder makes with the ground.
- Use tangent (rise/run):
[ \tan θ = \frac{3}{4} \quad (\text{because adjacent} = \sqrt{5^2-3^2}=4) ] - θ ≈ 36.9° (look up a table or use a calculator).
4. The Reciprocal Ratios
| Name | Reciprocal of | Formula |
|---|---|---|
| csc θ (cosecant) | sin θ | 1 / sin θ |
| sec θ (secant) | cos θ | 1 / cos θ |
| cot θ (cotangent) | tan θ | 1 / tan θ |
These are useful when you need the opposite side but only have the adjacent side or vice‑versa.
5. Quick Reference for Common Angles
| Angle | sin | cos | tan |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | ½ | √3/2 | 1/√3 |
| 45° | √2/2 | √2/2 | 1 |
| 60° | √3/2 | ½ | √3 |
| 90° | 1 | 0 | ∞ (undefined) |
(These numbers come from special 30‑60‑90 and 45‑45‑90 triangles.)
6. A Quick Practice Problem
Problem:
A flagpole is 12 m tall. A guy rope is attached at the top and meets the ground 16 m away from the pole base.
Find the angle the rope makes with the ground.
- Identify the right triangle:
- Opposite = 12 m (height)
- Adjacent = 16 m (run)
- Use tangent:
[ \tan θ = \frac{12}{16} = \frac{3}{4} ] - θ ≈ 36.9° (same as the ladder example).
7. Summary
- Trigonometric ratios are simple “divide‑your‑sides” rules for right‑angled triangles.
- Sine = opposite/hypotenuse – up/down.
- Cosine = adjacent/hypotenuse – side/back.
- Tangent = opposite/adjacent – steepness.
- The other three ratios are just the reciprocals.
- Memorize the values for 0°, 30°, 45°, 60°, 90° – they’re the building blocks for most problems.
Use these ratios whenever you need to find a missing side or angle in a right triangle, just like you’d use a simple recipe: mix the right ingredients (sides) and the ratio tells you the result (angle or side length). Happy calculating!