Geometry and Trigonometry

The three ratios — for a right-angled triangle with angle θ:

sinθ = opposite ÷ hypotenuse

cosθ = adjacent ÷ hypotenuse

tanθ = opposite ÷ adjacent

Pythagoras — c² = a² + b² for the hypotenuse c. Used constantly for the magnitude of a resultant of two perpendicular vectors.

Resolving a vector — a vector of magnitude F at angle θ has a component F cosθ adjacent to the angle and F sinθ opposite to it.

Do not memorise which is which — sketch the triangle and read it off. If the angle is measured from the vertical rather than the horizontal, sine and cosine swap, and a sketch catches that instantly while a memorised rule does not.

Example: a 20 N force at 30° above the horizontal has a horizontal component 20cos30° = 17.3 N and a vertical component 20sin30° = 10.0 N. Check it: 30° is closer to the horizontal, so the horizontal component should be the larger — and it is.

Finding an angle — use the inverse functions, sin⁻¹, cos⁻¹ and tan⁻¹, and make sure the calculator is in degrees unless the question uses radians. A calculator left in radians is a silent and very common source of wrong answers.

Radians — an alternative measure of angle, with 2π radians in a full turn, so

180° = π radians

Areas and volumes — circle area πr², circumference 2πr, sphere surface area 4πr², sphere volume (4/3)πr³, cylinder volume πr²h.

Working from diameter — measurements are usually of diameter, so halve it before using any formula that takes a radius. A cross-sectional area is πd²/4 if you prefer to work directly from the diameter.

Why the squares and cubes matter for uncertainty — since area depends on r², a percentage uncertainty in a radius doubles when it becomes the uncertainty in an area, and triples for a volume. This is why the diameter is usually the measurement most worth taking carefully.