A sector is a share of the circle

\text{share of the circle} = \frac{\theta}{2\pi}

s = \frac{\theta}{2\pi} \times 2\pi r = r\theta

A = \frac{\theta}{2\pi} \times \pi r^2 = \tfrac{1}{2}r^2\theta

in degrees the share would be θ/360, and a factor π/180 would follow you everywhere

Radians

Measure an angle with the circle's own radius.

Degrees split a turn into 360 for historical reasons. Radians use a length the circle already has, and that makes every formula for arcs, sectors and calculus simpler.

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One radian

Take a piece of string as long as the radius and bend it round the rim. The angle it makes at the centre is one radian, about $57.3^\circ$ , the same size in every circle.

Half a turn

Keep laying radius-lengths round the rim. Half a turn takes three and a little more: exactly $\pi$ of them, because half the circumference is $\pi r$ .

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

Arcs and sectors

An angle in radians counts radii along the rim, so the arc is simply the radius times the angle. In Example 8.5, $r = 6$ cm and $\theta = 1.2$ : the arc is $7.2$ cm and the area $21.6$ cm².

s = r\theta \qquad A = \tfrac{1}{2}r^2\theta

Small angles

Close up, a small angle's height $\sin\theta$ , its arc $\theta$ and its tangent $\tan\theta$ are almost the same length. At $\theta = 0.2$ , $\sin\theta = 0.1986\ldots$ and $\tan\theta = 0.2027\ldots$ So for small $\theta$ in radians, $\sin\theta \approx \theta$ and $\tan\theta \approx \theta$ .

Chapter 8

Count the angle in radii, and the formulas fall out.

Exact values, graphs and equations in radians, segments, the small-angle approximation for $\cos\theta$ , and twelve graded exam questions with full worked solutions are waiting.

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