Understanding Trigonometry: Sin, Cos & Tan Explained Simply

1. The Foundation: Right-Angled Triangles

Trigonometry is the study of relationships between side lengths and angles in triangles. For any right-angled triangle with reference acute angle θ:

• Hypotenuse (H): The longest side opposite the 90° right angle.
• Opposite (O): The side directly facing angle θ across the triangle.
• Adjacent (A): The side adjacent to angle θ that is not the hypotenuse.

2. SOH CAH TOA Memory Rule

The three fundamental trigonometric ratios are memorized via the classic mnemonic SOH CAH TOA:

• SOH: sin(θ) = Opposite / Hypotenuse
• CAH: cos(θ) = Adjacent / Hypotenuse
• TOA: tan(θ) = Opposite / Adjacent

3. Special Angles to Memorize

In academic exams and engineering tests, angles 30°, 45°, and 60° appear frequently:

• sin(30°) = 1/2 = 0.5, cos(30°) = √3/2 ≈ 0.866, tan(30°) = 1/√3 ≈ 0.577
• sin(45°) = √2/2 ≈ 0.707, cos(45°) = √2/2 ≈ 0.707, tan(45°) = 1.0
• sin(60°) = √3/2 ≈ 0.866, cos(60°) = 1/2 = 0.5, tan(60°) = √3 ≈ 1.732

4. Practical Worked Example: Finding Tree Height

Suppose you stand 20 meters away from a tall pine tree. Your inclinometer measures an angle of elevation of 35° from ground level to the treetop. How tall is the tree?

Here, the adjacent side is 20 m and we want the opposite height h:

tan(35°) = Opposite / Adjacent = h / 20

h = 20 × tan(35°) = 20 × 0.7002 = 14.004 meters