The idea

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.

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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

What a radian is

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

2\pi

times, because the circumference is

2\pi r

. So a full turn is

2\pi

radians, which gives the one fact everything else follows from:

\pi \text{ radians} = 180^\circ.

Converting

Multiply by

\dfrac{\pi}{180}

to go degrees

\to

radians, and by

\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^\circ = 30\times\frac{\pi}{180} = \frac{\pi}{6}, \qquad \frac{3\pi}{4} = \frac{3\pi}{4}\times\frac{180}{\pi} = 135^\circ.

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