Why Mathematicians Prefer Radians to Degrees for Measuring Angles

In earlier classes, an angle was a corner of a triangle, always less than $180^{\circ }$. Now an angle is the amount of rotation of a ray about a point, so it can be negative or **larger than $360^{\circ }$ — like a wheel that keeps turning.

Angles can also be measured in radians**, a unit based on the circle itself, which makes formulas for arcs and, later, calculus much simpler.

This part covers angles in standard position, degree–radian conversion and the formula $\theta =\frac{l}{r}$.

An angle is in standard position when its vertex is at the origin and its initial side lies along the positive x-axis; anticlockwise rotation gives a positive angle and clockwise rotation gives a negative angle.

Which quadrant?

- **$135^{\circ }$** — anticlockwise past $90^{\circ }$, so the terminal side is in quadrant II
- **$240^{\circ }$ — quadrant III
- $-60^{\circ }$ — clockwise from the x-axis, quadrant IV
- $-120^{\circ }$** — the same terminal side as $360^{\circ }-120^{\circ }=240^{\circ }$, quadrant III

Coterminal angles share a terminal side and differ by whole turns:

$$390^{\circ }=360^{\circ }+30^{\circ }\text{so }390^{\circ }\text{ and }30^{\circ }\text{ are coterminal}$$

An everyday example. The minute hand of a wall clock moves clockwise, so in $15$ minutes it turns through $-90^{\circ }$ in standard position.

The substance. **Angles beyond $360^{\circ }$ are meaningful** — a ceiling fan blade turning twice and a quarter turns through $810^{\circ }$.

**One radian is the angle subtended at the centre by an arc equal in length to the radius; since a full turn is $2\pi$ radians, $\pi$ radians $=180^{\circ }$, so multiply degrees by $\frac{\pi }{180}$ to get radians and radians by $\frac{180}{\pi }$ to get degrees.**

$$\pi \text{ rad}=180^{\circ }{1}^{\circ }=\frac{\pi }{180}\text{ rad}\approx 0.01745\text{ rad}1\text{ rad}=\frac{180^{\circ }}{\pi }\approx 57.3^{\circ }$$

Degrees to radians.

$$45^{\circ }=\frac{\pi }{4}240^{\circ }=240\times \frac{\pi }{180}=\frac{4\pi }{3}-47.5^{\circ }=-47.5\times \frac{\pi }{180}=-\frac{19\pi }{72}$$

Radians to degrees.

$$\frac{5\pi }{3}=\frac{5\times 180^{\circ }}{3}=300^{\circ }2\text{ rad}=\frac{360^{\circ }}{\pi }\approx 114.6^{\circ }$$

An everyday example. A ceiling fan blade that makes half a turn has rotated through $\pi$ radians, whatever the length of the blade.

The substance. A radian is a ratio of two lengths, so it has no physical unit — which is why calculators need a separate RAD mode.

**An arc of length $l$ on a circle of radius $r$ subtends an angle $\theta =\frac{l}{r}$ radians at the centre, so $l=r\theta$ and $r=\frac{l}{\theta }$, with $\theta$ always in radians.

Worked example 1 — arc length.** $r=21$ cm, $\theta =60^{\circ }=\frac{\pi }{3}$.

$$l=21\times \frac{\pi }{3}=7\pi =7\times \frac{22}{7}=22\text{ cm}$$

Worked example 2 — angle. A pendulum $75$ cm long swings so that its tip moves along an arc of $21$ cm.

$$\theta =\frac{21}{75}=\frac{7}{25}\text{ rad}$$

Worked example 3 — clock. A minute hand $1.5$ cm long moves for $40$ minutes, turning $240^{\circ }=\frac{4\pi }{3}$.

$$l=1.5\times \frac{4\pi }{3}=2\pi \approx 6.28\text{ cm}$$

Worked example 4 — radius. An arc of $37.4$ cm subtends $60^{\circ }$.

$$r=\frac{37.4}{\pi /3}=\frac{37.4\times 3\times 7}{22}=35.7\text{ cm}$$

An everyday example. **A train runs at $66$ km/h on a curve of radius $1500$ m.** In $10$ s it covers $\frac{66000}{3600}\times 10\approx 183.3$ m, turning through

$$\theta =\frac{183.3}{1500}\approx 0.122\text{ rad}\approx 7^{\circ }$$

The trap. **Using degrees in $l=r\theta$** gives an arc $60$ times too long for $60^{\circ }$.