Compare an arc with its radius

If an arc has length $s$ in a circle of radius $R>0$, its measure in radians is:

$$\theta=\frac{s}{R}.$$

A radian is the central angle that cuts an arc as long as the radius. Since the length of a circle is $2\pi R$, one revolution measures $2\pi$ radians. It follows that:

$$360°=2\pi\text{ rad},\qquad180°=\pi\text{ rad}.$$

The conversions are $\theta_{\mathrm{rad}}=\theta_{\mathrm{graus}}\pi/180$ and $\theta_{\mathrm{graus}}=\theta_{\mathrm{rad}}180/\pi$. For example, $75°=5\pi/12$ rad.

Demonstrate the length formula

An arc of $\alpha$ degrees occupies fraction $\alpha/360$ of the turn. Its length is:

$$s=\frac{\alpha}{360}\,2\pi R=R\frac{\alpha\pi}{180}=R\theta.$$

The expression $s=R\theta$ requires $\theta$ to be in radians. For an arc of radius 8 cm and measurement $\pi/3$, $s=8\pi/3$ cm. For length we use the length covered, which is non-negative; an oriented angular displacement can have sign.

The number line goes around

The trigonometric cycle is the circle with radius 1 and center at the origin. We associate the real number $x$ with the point reached from $(1,0)$ after a rotation of $x$ radians. The positive direction is counterclockwise; the negative, time.

The first quadrant corresponds to $0

Two reals have the same image exactly when their difference is an integer number of turns:

$$x\equiv y\pmod{2\pi}\iff x-y=2k\pi,\qquad k\in\mathbb Z.$$

Thus, we can write $x=x_0+2k\pi$ with $0\le x_0<2\pi$. This representation separates the completed turns from the final position.

Example with a negative spin

For $x=-7\pi/6$, we add one turn: $-7\pi/6+2\pi=5\pi/6$. The final position is in the second quadrant, although the rotation was carried out in the negative direction.

In everyday life

  • Car wheels. At 72 km/h (20 m/s), a wheel with radius 0,3 m rotates $\frac{20}{0{,}3} \approx 66{,}7$ rad/s, about 637 revolutions per minute — the formula $v = \omega r$ is only valid with the angle in radians.
  • Athletics tracks. Each curve is half a circle; with a radius of 36,5 m, it measures around $\pi \cdot 36{,}5 \approx 115$ m.
  • Programming. The functions sin and cos in programming languages ​​and spreadsheets receive radians; Forgetting the conversion is one of the most common mistakes.
Central angle in degrees and radians, and the corresponding arc: $\ell = r\theta$ only applies to radians.

Exercises with resolution

1. Convert $225°$ to radians and $7\pi/6$ rad to degrees.

Check resolution 1

$225\pi/180=5\pi/4$ rad; $(7\pi/6)180/\pi=210°$.

2. A circle has a radius of 3 cm. What is the length of an arc of $120°$?

Check resolution 2

$120°=2\pi/3$ rad. Then $s=3\cdot2\pi/3=2\pi$ cm.

In the interactive book

The same subjects, with all the demonstrations and the complete list of exercises, are in the interactive Trigonometry book:

This class is a shorter introduction to the topic; In the book, the theory is accompanied by solved and proposed exercises from the chapter.

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