Radians

A radian is the angle subtended when the arc length equals the radius, which is exactly why $s = r\theta$ and $A = \tfrac{1}{2}r^2\theta$ only work in radians. One conversion fact drives everything:

$\pi$ radians $= 180°$

Converting both ways

• Degrees $\to$ radians: $\times \frac{\pi}{180}$. So $60° = \frac{\pi}{3}$, $150° = \frac{5\pi}{6}$, $270° = \frac{3\pi}{2}$
• Radians $\to$ degrees: $\times \frac{180}{\pi}$. So $0.8$ rad $= \frac{144}{\pi} \approx 45.8°$

On Paper 1, radian answers stay as exact multiples of $\pi$, $\frac{\pi}{3}$, not $1.05$. Learn the standard set as a table you can write from memory: $30° = \frac{\pi}{6}$, $45° = \frac{\pi}{4}$, $60° = \frac{\pi}{3}$, $90° = \frac{\pi}{2}$, $180° = \pi$, $360° = 2\pi$. These pair with the exact trig values: $\sin(\pi/6) = \tfrac{1}{2}$ must be as instant as $\sin 30° = \tfrac{1}{2}$.

Reading the units before anything else

The highest-value habit in this corner of the syllabus: circle the angle units (or range) in the question before computing. A trig equation with range $0 \le x \le 2\pi$ demands radian answers; an arc-length formula fed degrees produces garbage off by a factor of $\approx 57.3$; a calculator in degree mode evaluating $\sin 1.2$ (radians intended) produces a wrong-but-plausible number, the worst kind. Mode discipline is method: setting radian mode is part of the working on Paper 2, and unit errors head the examiner-report repeat list.

Why 0606 prefers radians

Beyond circular measure itself, radians are the native units of trig graphs and equations with $\pi$-ranges, and of all trigonometric calculus, $\frac{d}{dx}(\sin x) = \cos x$ is only true in radians. The syllabus expects you to live comfortably in both unit systems and to detect which one a question speaks.

Common mistakes

• Conversion factor inverted (multiplying by $\frac{180}{\pi}$ when degrees $\to$ radians)
• Exact-$\pi$ answers decimalised on Paper 1
• Calculator mode unchecked across a paper that mixes units between questions
• Range $0$ to $2\pi$ answered in degrees (or vice versa)
• Treating “1 radian” as a strange unit rather than $\approx 57.3°$, a sense-check anchor worth keeping

Full topic context: Circular Measure notes.