Angles and their measure: degrees and radians

Oriented angles

In elementary geometry, an angle is the figure formed by two rays with a common endpoint, and its measure lies between $0^\circ$ and $180^\circ$. To study rotations, such as that of a wheel, of the hands of a clock or of the Earth about its axis, this notion is not enough: what matters is how far something has turned and in which direction, and a rotation may make several turns. For this reason the angle is regarded as the result of a rotation.

An oriented angle is determined by a ray, called the initial side, which turns about its endpoint, called the vertex, until it reaches a final position, called the terminal side, together with the rotation performed. The angle is positive if the rotation is made in the direction opposite to that of the hands of a clock (counterclockwise), and negative if it is made in the direction of the hands of a clock (clockwise). The zero angle is the one that does not turn.

One and the same pair of sides may correspond to many oriented angles: the terminal side reaches the same position if it turns a quarter turn counterclockwise, three quarters of a turn clockwise, or a quarter turn plus a full turn. These angles are different, because they describe different rotations, even though they end on the same side; the third section makes precise how they are related.

The sexagesimal system

The oldest measure of angles divides the full turn into $360$ equal arcs, and calls degree the angle that intercepts each of them: an angle measures $g$ degrees if its arc is $g$ times the arc of one degree, so that the measure in degrees is proportional to the length of the arc. Each degree is divided into $60$ minutes, and each minute into $60$ seconds. The system comes from Babylonian astronomy, which counted in base sixty, as explained in the Course I lesson on positional numeral systems; the number $360$ has the advantage of being divisible by $2$, $3$, $4$, $5$, $6$, $8$, $9$, $10$ and $12$, among others. A right angle measures $90^\circ$, a straight angle measures $180^\circ$ and a full turn, $360^\circ$; a negative angle is measured with the minus sign, so that a quarter turn clockwise measures $-90^\circ$. The angle of 40 degrees, 15 minutes and 36 seconds, which is abbreviated by writing after each figure the degree sign, one prime for the minutes and two for the seconds, measures the following in degrees, that is, 40.26 degrees:

The number $360$ is, however, a convention: nothing in the geometry of the circle distinguishes it from any other number. The formulas for the length of an arc and the area of a sector, and later those of the trigonometric functions, become simpler if angles are measured in a way that arises from the circle itself.

The radian

The idea is to measure an angle by the length of the arc it intercepts on a circle of radius $1$ centered at its vertex. This requires knowing what the length of an arc is, and here a difficulty appears: an arc is not a union of line segments, and its length is defined as the value approached by the lengths of the polygonal lines inscribed in it as their sides become shorter and shorter, that is, by means of a limit. Its study belongs to the course Calculus II, where lengths of curves and areas of regions bounded by curves are computed. Here the following facts are admitted.

Admitted fact (lengths and areas in the circle). Every arc of a circle has a length, and every region bounded by line segments and arcs has an area. Congruent figures have the same length or the same area; if a figure is decomposed into two that do not overlap, its measure is the sum of the measures of the pieces; and if one region contains another, its area is not smaller, and it is larger if the difference contains some disk. Moreover, if two circles have radii $r$ and $R$, the arcs that one and the same central angle intercepts on them are proportional to the radii, and the area of a disk of radius $r$ is $\pi r^2$.

This fact is admitted without proof, because the lengths of arcs and the areas of curved regions are defined by means of limits and computed with the integral, in the course Calculus II. Its last clause presupposes the number $\pi$, which is now defined: the number $\pi$ is half the circumference of the unit circle, that is, of the circle of radius $1$. By the proportionality of arcs to radii, applied to the angle of a full turn, the circle of radius $r$ has circumference $2\pi r$, and the ratio of the circumference of any circle to its diameter is always $\pi$. It is also admitted that $\pi$ is irrational, and that its decimal expansion begins with 3.14159; in particular, $3 < \pi < 4$.

To measure angles one must also traverse the circle by arc length: start from a point and advance along it a given length, in one direction or the other, and know which point is reached. Everything this unit says about angles and trigonometric functions rests on that traversal, so it is worth stating precisely what is admitted about it. It is stated in the Cartesian plane, for the unit circle $x^2 + y^2 = 1$ and its point $A = (1, 0)$; since lengths do not change when figures are moved, it holds equally for any circle of radius $1$ and any starting point.

Axiom (traversal by arc length). For each real number $s$ there is a unique point $P(s)$ of the unit circle which is reached by starting from $A = (1, 0)$ and traversing along it a length $\abs{s}$, counterclockwise if $s > 0$ and clockwise if $s < 0$, making as many turns as necessary; for $s = 0$ we have $P(0) = A$. This traversal has the four properties below.

1. Concatenation: if from the point $P(s)$ one traverses a length $t \geq 0$ counterclockwise, one reaches $P(s + t)$; if one traverses it clockwise, one reaches $P(s - t)$.
2. Periodicity: a full turn, in either direction, traverses the length $2\pi$ of the circle and returns to the starting point; that is, $P(s + 2\pi) = P(s)$ for every real number $s$.
3. Half turn: the counterclockwise traversal first passes through the upper semicircle, from right to left. Precisely, $P(\pi) = (-1, 0)$, and if $0 \leq s < t \leq \pi$, the points $P(s)$ and $P(t)$ have nonnegative ordinate and the abscissa of $P(s)$ is greater than that of $P(t)$.
4. Covering: every point of the unit circle is $P(s)$ for some real number $s$.

The axiom is admitted without proof for the same reason as the fact stated before it: the length of an arc is defined by means of a limit, and its properties are proved in the course Calculus II; covering, which asserts that in traversing the circle no point is skipped, rests moreover on the intermediate value property of continuous functions, which is proved in the course Calculus I. The half-turn property has a simple geometric reading. If $(x, y)$ lies on the unit circle, the square of its distance to the point $A$ is as follows:

Hence saying that the abscissa decreases strictly along the upper semicircle is equivalent to saying that the chord joining $A$ to the point reached increases strictly with the arc, until it becomes a diameter after a half turn. When, in the lesson on trigonometric ratios, the cosine of $s$ is defined as the abscissa of $P(s)$, this property will say that the cosine is strictly decreasing on $\icc{0}{\pi}$: that monotonicity is not deduced; it is the half-turn clause itself read in coordinates. Now, from this axiom, from the facts about lengths and areas and from the fact about reflections admitted below, everything else that is used about the traversal is deduced: in this lesson, the behavior of the second half turn and the fact that one turn does not pass twice through the same point; in the lesson on the inverse trigonometric functions, the monotonicity of the sine and of the tangent.

Admitted fact (reflections and the half-turn rotation). A reflection in a line preserves distances and, with them, the lengths of arcs, but it reverses the direction of rotation: it transforms a counterclockwise traversal of a circle centered at a point of that line into a clockwise traversal of the same length, and conversely. The half-turn rotation about a point, which is the composition of the reflections in two perpendicular lines through it, preserves distances, lengths and also the direction of rotation.

This fact belongs to the elementary geometry of plane motions, which this course does not develop. In coordinates, the reflection in the $x$-axis is the transformation $(x, y) \mapsto (x, -y)$; it maps the unit circle onto itself, because $x^2 + (-y)^2 = x^2 + y^2$, and leaves the point $A = (1, 0)$ fixed. With it, the second half turn of the traversal is obtained from the first.

Consider an oriented angle with vertex at the center $O$ of the unit circle, and let $A$ be the point where the initial side meets that circle. As the side turns, the point $A$ traverses an arc of the circle, as in the axiom stated above. The radian measure of the angle is the length of that arc, with positive sign if the rotation is counterclockwise and negative if it is clockwise; if the rotation makes several turns, each full turn contributes the length $2\pi$ of the circle.

An angle measured in radians is a dimensionless real number: it is the ratio of an arc length to the length of the radius, which equals $1$. That is why one simply writes $\theta = 2$ for an angle of two radians; when one wishes to stress the unit, one writes 2 rad. By contrast, a measure in degrees is always indicated with its symbol, as in $30^\circ$. Conversely, every real number is the measure of some oriented angle: by the axiom of traversal, for each $\theta$ the point $A$ can traverse the circle a length $\abs{\theta}$ in the direction indicated by the sign of $\theta$, making as many full turns as necessary, and the ray from the vertex to the point reached is the terminal side.

Converting between degrees and radians

Since the full turn measures $360^\circ$ and also $2\pi$ radians, and since both measures are proportional to the length of the arc that the angle intercepts on the unit circle (the degree measure by the definition of the degree, and the radian measure because it is that length), the two measures of an angle are proportional.

In practice, to pass from degrees to radians one multiplies by $\frac{\pi}{180}$, and to pass from radians to degrees, by $\frac{180}{\pi}$. Thus $225^\circ = \frac{5\pi}{4}$, $-60^\circ = -\frac{\pi}{3}$ and $\frac{7\pi}{6} = 210^\circ$; the angle of $2$ radians measures $\frac{360}{\pi}$ degrees, approximately 114.6 degrees.

Arc length and area of a sector

On a circle of radius $r$, a central angle $\theta$ intercepts an arc, and the two sides of the angle, together with that arc, bound a circular sector. Radian measure gives their formulas a particularly simple form.

If the angle were measured in degrees, the formulas would carry the conversion factor: the arc would be $\frac{\pi r g}{180}$ and the area of the sector $\frac{\pi r^2 g}{360}$. The simplicity of $s = r\theta$ is the main reason for preferring the radian. Moreover, the formula gives the measure of an angle on any circle, not only on the unit circle: $\theta = \frac{s}{r}$ is the ratio of the arc to the radius, and it does not depend on the radius chosen.

The inscribed angle

An angle with its vertex at the center of a circle is called a central angle. If $A$, $B$ and $C$ are three distinct points of a circle, the angle formed at $C$ by the chords $CA$ and $CB$ is called an inscribed angle, and it is said to intercept the arc with endpoints $A$ and $B$ that does not contain $C$. By the theorem proved above, to an arc of length $s$ on a circle of radius $r$ corresponds the central angle of $\frac{s}{r}$ radians, even if that number exceeds $\pi$. To compare the inscribed angle with the central one, the following facts of elementary geometry are admitted, which this course does not prove because it does not develop synthetic geometry. The first two measure the angles of a triangle; the last two are facts of position, which say how angles and arcs are shared out when a ray passes through the interior of an angle.

Admitted fact (angles of the triangle and positions on the circle). The four statements below are admitted, in which non-oriented angles are measured in radians, between $0$ and $\pi$.

1. Sum of the angles: the three angles of a triangle add up to a straight angle, that is, $\pi$ radians.
2. Isosceles triangle: in a triangle with two equal sides, the angles opposite those sides are equal.
3. Sum of adjacent angles: if a ray starting from the vertex of an angle passes through its interior, it divides it into two angles whose measures add up to that of the given angle.
4. Position of the center: let $A$, $B$ and $C$ be three distinct points of a circle with center $O$, and let $D$ be the other endpoint of the diameter starting from $C$. If $O$ lies on neither of the chords $CA$ and $CB$, one of two things happens. Either $O$ lies in the interior of the angle formed at $C$ by those chords, and then the ray from $C$ to $D$ passes through that interior and $D$ lies on the arc with endpoints $A$ and $B$ that does not contain $C$. Or $O$ lies outside that angle, and then one of the points $A$ and $B$, which may be called $A$, lies on the arc with endpoints $D$ and $B$ that does not contain $C$, and the ray from $C$ to $A$ passes through the interior of the angle formed at $C$ by the chords $CD$ and $CB$.

Points that do not lie on the circle see the chord under other angles. The following result is also admitted from elementary geometry: it follows from the theorem above and from proposition 21 of book I of Euclid's Elements, according to which a point inside a triangle sees one of its sides under an angle greater than the opposite vertex does; but its proof requires knowing where the line segments starting from the point meet the circle, a question of position that this course does not develop. It will be used in the last solved problem of the lesson on the inverse trigonometric functions.

Admitted fact (angles from interior and exterior points). Let $A$ and $B$ be two points of a circle, and $\gamma$ the common measure of the inscribed angles with endpoints $A$ and $B$ whose vertex lies on one of the two arcs these points determine. If $Q$ is a point not on the circle, situated on the same side of the line $AB$ as that arc, the angle formed at $Q$ by the line segments $QA$ and $QB$ is less than $\gamma$ if $Q$ is outside the disk, and greater than $\gamma$ if it is inside.

Standard position and coterminal angles

In the Cartesian plane, studied in the Course I lesson on the equation of a line and Cartesian coordinate systems, every angle has a privileged position.

An oriented angle is in standard position if its vertex is the origin and its initial side is the positive $x$-semi-axis. Its terminal side is then the ray that starts at the origin and passes through the point $P(\theta)$ of the axiom of traversal, that is, the point of the unit circle $x^2 + y^2 = 1$ reached by traversing, from $(1, 0)$, an arc of length $\abs{\theta}$ in the direction indicated by the sign of $\theta$. Two angles are coterminal if, in standard position, they have the same terminal side.

The axes divide the plane into four quadrants, numbered counterclockwise starting from the one that contains the points with positive coordinates. An angle in standard position lies in the quadrant that contains its terminal side; if that side falls on an axis, the angle is called quadrantal. The coordinates of the point $P(\theta)$ are the cosine and the sine of $\theta$, which will be studied in the lesson that follows this one, on trigonometric ratios.

A full turn, in either direction, returns the side to its position, by the periodicity of the traversal: the angle $\theta + 2\pi$ is coterminal with $\theta$, and so are $\theta - 2\pi$, $\theta + 4\pi$ and, in general, $\theta + 2k\pi$ with $k$ an integer. To know where an arbitrary angle ends, then, it suffices to discount the full turns, as the multiples of the divisor are discounted in the Course I lesson on divisibility and division with remainder. The theorem below is the real version of that division: the divisor is $2\pi$, the quotient is the number of turns and the remainder, the angle that is left.

The number $\theta_0$ is called the reduction of $\theta$ to the first turn. In degrees, the same reasoning, with $360$ instead of $2\pi$, reduces any angle to one between $0^\circ$ and $360^\circ$, the latter excluded: for example, $1000^\circ = 280^\circ + 2 \cdot 360^\circ$. With the reduction one recognizes all the angles that end on the same side; but first one needs to know that one turn does not pass twice through the same point, which is deduced from the axiom of traversal.

It follows that the angles coterminal with $\theta$ are exactly those of the form $\theta + 2k\pi$, with $k$ an integer, and that the quadrant of an angle is decided by its reduction: it lies in the first quadrant if $0 < \theta_0 < \frac{\pi}{2}$, in the second if $\frac{\pi}{2} < \theta_0 < \pi$, in the third if $\pi < \theta_0 < \frac{3\pi}{2}$ and in the fourth if $\frac{3\pi}{2} < \theta_0 < 2\pi$; it is quadrantal if $\theta_0$ is $0$, $\frac{\pi}{2}$, $\pi$ or $\frac{3\pi}{2}$. Thus $-\frac{23\pi}{6}$ lies in the first quadrant, and $1000^\circ$, whose reduction is $280^\circ$, in the fourth.

Finally, every point of the unit circle is the point $P(\theta)$ of some angle: this is the covering property of the axiom of traversal, which says that in traversing the circle from $(1, 0)$ one passes through all its points. If this is so, by the reduction to the first turn, periodicity and the lemma on coterminal angles, each point of the unit circle is $P(\theta)$ for a unique $\theta$ in $\ico{0}{2\pi}$, and the angles corresponding to it are exactly those coterminal with that $\theta$.

Angular velocity and linear velocity

When a body rotates about an axis, all its points describe arcs of circles centered on the axis, and all of them turn through the same angle in the same time. If in a time $t$ the body turns, at a constant rate, through an angle $\theta$, its angular velocity is the ratio $\omega = \frac{\theta}{t}$, which is measured in radians per second; it is also expressed in revolutions per minute, and since one revolution is $2\pi$ radians and one minute, $60$ seconds, one revolution per minute is equivalent to $\frac{\pi}{30}$ radians per second.

A point at distance $r$ from the axis traverses, in that time, an arc of length $s = r\theta$, by the arc length theorem. Its linear velocity, the ratio of the distance traveled to the time, is therefore the product of the radius and the angular velocity:

It follows that, in a rotating body, the angular velocity is the same for all points, whereas the linear velocity grows in proportion to the distance from the axis: the rim of a spinning disk moves faster than the points near the center. This relation, valid only if angles are measured in radians, is the starting point of the study of circular motion in the course Foundations of Mechanics. Two wheels joined by a belt or a chain that does not slip, or that mesh with each other, have the same linear velocity at the point of contact; therefore $\omega_1 r_1 = \omega_2 r_2$, and the smaller wheel turns faster, as will be seen in solved problem 3.

Solved problems

The problems measure the Earth from an angle, compute the area of a circular segment, transmit a rotation through a chain, look for the sector of greatest area with a given perimeter and count the overlaps of the hands of a clock. Each one calls for an idea of its own, which is worth looking for before reading the solution.

Eratosthenes expressed the distance in stadia, whose exact value is disputed; with the figure adopted here, the meridian measures 40000 km, and not by chance: at the end of the eighteenth century the meter was defined as the ten-millionth part of the quarter meridian. On the other hand, Alexandria is not exactly north of Syene but some three degrees farther west, so the arc joining them is not quite an arc of meridian.