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Pure · Trigonometry

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Radian measure

What a radian is, converting between degrees and radians, the exact values at the special angles in radians, and solving trigonometric equations over an interval given in radians.

Pure · Trigonometry

Radian measure

What a radian is, converting between degrees and radians, the exact values at the special angles in radians, and solving trigonometric equations over an interval given in radians.

Why it works

A radian is just a different unit for measuring an angle — and a more natural one. One radian is the angle subtended at the centre of a circle by an arc whose length equals the radius. Wrap the radius around the circumference: it goes round 2π2\pi times, because the circumference is 2πr2\pi r. So a full turn is 2π2\pi radians, which gives the one fact everything else follows from: π radians=180∘.\pi \text{ radians} = 180^\circ.

Converting. Multiply by π180\dfrac{\pi}{180} to go degrees →\to radians, and by 180π\dfrac{180}{\pi} to go radians →\to degrees. (They're reciprocals, so if you ever forget which way, check that the units you don't want cancel.)

30∘=30×π180=π6,3π4=3π4×180π=135∘.30^\circ = 30\times\frac{\pi}{180} = \frac{\pi}{6}, \qquad \frac{3\pi}{4} = \frac{3\pi}{4}\times\frac{180}{\pi} = 135^\circ.

Why bother? Radians make the rest of A-level work. The arc-length and sector formulas become simply s=rθs = r\theta and A=12r2θA = \tfrac12 r^2\theta (no clumsy θ360\frac{\theta}{360} factor), and the calculus of trig functions — ddxsin⁡x=cos⁡x\frac{d}{dx}\sin x = \cos x — is only true when xx is in radians. From now on, assume radians unless a question states degrees, and set your calculator accordingly.

The special angles, in radians. The exact values you learned at AS don't change — only the label on the angle does: θπ6π4π3π2sin⁡θ1222321cos⁡θ3222120\begin{array}{c|cccc} \theta & \dfrac{\pi}{6} & \dfrac{\pi}{4} & \dfrac{\pi}{3} & \dfrac{\pi}{2}\\[4pt] \hline \sin\theta & \tfrac12 & \tfrac{\sqrt2}{2} & \tfrac{\sqrt3}{2} & 1\\[4pt] \cos\theta & \tfrac{\sqrt3}{2} & \tfrac{\sqrt2}{2} & \tfrac12 & 0 \end{array} (π6=30∘\frac{\pi}{6}=30^\circ, π4=45∘\frac{\pi}{4}=45^\circ, π3=60∘\frac{\pi}{3}=60^\circ.)

Solving equations in radians. The method is exactly as before — find the principal value, then use the symmetry of the graph (or the CAST quadrants) to get every solution in the interval. The only differences: the interval is written in radians (often 0≤x≤2π0 \le x \le 2\pi), and you give exact answers as multiples of π\pi whenever the angle is special. For cos⁡x=k\cos x = k the solutions in [0,2π][0,2\pi] are x=αx = \alpha and x=2π−αx = 2\pi - \alpha; for sin⁡x=k\sin x = k they are x=αx=\alpha and x=π−αx=\pi-\alpha. If the equation involves 2x2x or x−π6x-\tfrac{\pi}{6}, transform the interval first, solve for that whole expression, then convert back.