Question 1

Why does maths use radians instead of degrees?

Accepted Answer

The formulas come out cleaner: arc length = r × θ and sector area = r²θ/2 only work that way in radians. The derivative of sin(x) is cos(x) only in radians — in degrees it gains a factor of π/180. Physics, engineering, and programming all use radians by default.

Question 2

What are arcminutes and arcseconds?

Accepted Answer

One degree splits into 60 arcminutes (′), and each arcminute into 60 arcseconds (″), so 1° = 3,600″. They matter in navigation (GPS precision is roughly 1″ ≈ 30 m), astronomy (the Moon spans about 30′), and optics. One arcsecond is about 5 mm at a distance of 1 km.

Question 3

What is a milliradian in shooting?

Accepted Answer

1 mil = 0.001 rad ≈ 0.0573°. Conveniently, at 1,000 m one mil subtends exactly 1 m. Rifle scopes are marked in mils ("come up 2 mils" = 2 m at 1 km) or in MOA (arcminutes), where 1 MOA ≈ 2.91 cm at 100 m.

Question 4

How fast does an engine spin in radians?

Accepted Answer

Engine speed is usually given in RPM (revolutions per minute). 3,000 RPM = 50 rev/s = 50 × 2π ≈ 314.16 rad/s = 18,000°/s. A slow diesel runs 1,000–2,500 RPM, a petrol engine 800–7,000 RPM, and a jet turbine 10,000–30,000 RPM.

Question 5

How are angles used in GPS?

Accepted Answer

Coordinates are latitude and longitude in degrees, minutes, and seconds. One degree of latitude is about 111 km, one arcminute about 1.85 km (a nautical mile), and one arcsecond about 30 m. A location can be written 55°45′07″N 37°37′04″E or as decimal degrees 55.7519°N, 37.6176°E.

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