Radians (Arc Length & Sector Area)

Why This Matters

At GCSE you measured angles in degrees. At A-Level, you switch to radians — and for good reason.

Here is the key result that motivates the entire switch. When $\theta$ is measured in radians:

$\lim_{\theta \to 0} \frac{\sin\theta}{\theta} = 1$

This limit is only true in radians. In degrees, $\lim_{\theta \to 0} \frac{\sin\theta°}{\theta} = \frac{\pi}{180} \approx 0.01745$. This limit is the reason that $\frac{d}{dx}(\sin x) = \cos x$ — only when $x$ is in radians. If you used degrees, every derivative and integral involving trig would carry an extra factor of $\frac{\pi}{180}$. Radians eliminate this entirely.

Beyond calculus, radians make the formulae for arc length and sector area far simpler. Radians are not just an alternative to degrees — they are the natural unit of angle measurement, defined directly from the geometry of the circle. Every A-Level topic from here on — trig equations, calculus, series — assumes you are working in radians.