The system of angular measurement now to be described, is sometimes referred to as the theoretical system of measurement. In this system the unit angle is the angle which at the centre of a circle subtends an arc equal in length to the radius.. This unit angle is called a radian. Thus, if a circle with any radius be described about O as a centre, and an arc AB be taken equal in length to the radius, then the angle AOB is a radian.

72. The value of a radian. In order that a quantity may be used as a unit of measurement, it must have a fixed value; that is, using the customary mathematical phrase, it must be a constant quantity. The proof that a radian has a fixed value, or is a constant quantity, depends upon two geometrical facts, viz.:

(a) In the same circle two angles at the centre are in the same ratio as their intercepted arcs.

(b) The ratio of a circumference of a circle to its diameter is the same for all circles. [See Art. 9 (b).]

For the proof of (a), reference may be made to any plane geometry; for instance, to Euclid VI., 33.* The proof of (b) is not contained in all geometries; for instance, Euclid does not give it.† Accordingly, an outline of such a proof and the calculation of are given in Note C of the Appendix. This note should now be studied by those whose course in plane geometry has not included

*The truth of theorem (a) can easily, by an inductive method, be made evident to students who have not proved the theorem in plane geometry. Thus, on taking angles which are twice, three times, four times, one-half, onethird, etc., of a given angle, it can be seen that their respective arcs bear the same relations to one another.

† Euclid lived about 323-283 B.C. Archimedes (287 ?-212 B.C.), the greatest mathematician of antiquity, measured the length of the circle and · the area contained by it, and also measured the surface of the sphere. He showed that the ratio of the circle to its diameter lies between 223 and 24. In 1794 a French mathematician, Adrien Marie Legendre (1752–1833), published his Elements of Geometry, in which the works of Euclid and Archimedes on elementary geometry are blended together. The elementary textbooks now in use on the continent of Europe and in the United States, are written mainly on Legendrean lines; the geometrical text-books generally studied throughout the British Empire, are editions of Euclid's Elements.

the measurement of the circle. Theorems (a) and (b) are assumed

Since all right angles are equal, and since each radian is a fixed

fraction, namely, 2, of a right angle, it follows that all radians

π

are equal. It will be remembered that the unit in the common practical system is one-ninetieth of a right angle.

Ex. With a protractor lay off an angle approximately equal to a radian. Compare it with angle 60°. An angle 60°, at centre of a circle, is subtended by a chord equal in length to the radius; a radian is subtended by an arc equal in length to the radius.

73. The radian measure of an angle. Measure of a circular arc. The radian measure of an angle is the ratio of the angle to a radian. [See Art. 8.] For instance, if an angle A is twice a radian, then its radian measure is 2; if an angle B is two-thirds of a radian, then its radian measure is . This is expressed thus:

O

(3)

Here, r is used as the symbol for radians just as is used as the symbol for degrees in 23°. In general discussions the radian

*The value of the radian has been calculated by J. W. L. Glaisher to 41 -places of decimals of a second. [Proc. Lond. Math. Soc., Vol. IV. (1871–73), pp. 308-312.]

measure of an angle is often expressed by Greek letters; thus, the angles a, ß, 0, &, etc., contain a, ß, 0, 4, etc., radians. In these cases the symbol is usually omitted, but it is always understood that the radian is the unit of measurement.

If the circular arc subtended by an angle is equal in length to twice the radius, then the radian measure of the angle is obviously two; if the arc is one-half the length of the radius, then the angle contains half a radian. The radian measure of an angle may be given a second definition, which depends on Theorem (a), Art. 72. Let AOP, Fig. 66, be any angle, and AOB be a radian. Describe a circle with any radius OA, equal to r, about the vertex O as a centre. Let arc AB be equal to the radius, and draw OB. Then angle AOB is a radian, by the definition in Art. 71.

That is, the number of radians in an angle, or the radian measure of an angle, is the answer to the question: how many times does any circular arc subtended by it, contain the radius? Thus, for example, the radian measures of the angles which subtend circular arcs equal in length to 2, 3, 1.5, .825 radii are 2, 3, 1.5, .825, respectively.

The circular arc subtended by 360° = 2 πr;

This shows that an angle 2π radians is described each time that the revolving line makes a complete revolution. Relation (6), namely,

connects the two systems of angular measurement. By means of (7), an angle expressed in the one system can be expressed in the other. The word radians is usually omitted from (7), but is always understood. Relation (7) may also be deduced directly

73.]

RADIAN MEASURE OF AN ANGLE.

125

from (1). Just as angles are considered as unlimited in magnitude, so arcs are considered as unlimited in length.

NOTE 1. The term circular measure is often used for radian measure, and c is used as the symbol for radians. Thus (3) is written A = 2o, B = }o.

The term radians and the symbol for radian is usually omitted from the second members of these equations, but is always understood to be there.

Ex. 2. Express 45°, 60°, 135°, 210°, 300°, 330°, 270°, 225°, — 75°, 63°, 27°, – 150°, in radian measure, (a) as fractions of π, (b) numerically, on putting = 24.

33°,

6 π, 1⁄2 π, § π, in degrees, and their complements and supplements, in radians.

Ex. 5. Express the angles - π, − 5 π, — § π, — 11 π, — 25 π,

Ex. 6. Express 2o (2 radians) in degrees.

in degrees.

then length of arc = radius × number of radians in the angle.

If a denote the length of any arc AP, r the radius, ✪ the radian measure of angle AOP, then

a = ro.

(8)

In words: The length of any circular arc is equal to the product of the radius and the radian measure of its subtended central angle. For example, the arc of 360° = 2π radii, arc of 180° =π radii, etc. These arcs are usually referred to as the arcs 2π, π, etc.; but it is always understood that the radius is the unit of measurement. The symbol, which always denotes the incommensurable number 3.14159 can thus be used in three connections in trigonometry: .

(1) With other numbers, as a number simply.

(2) With reference to angles; in which case it denotes an angle containing radians, i.e. 3.14159 ... radians.

(3) With reference to arcs; in which case it denotes an arc containing 3.14159 radii. This is an arc subtended by a central angle of radians.

The expression 180° = does not mean 180° it means 180°

=

3.14159 radians.

=

3.14159

...

The expression "arc π "does not mean arc 3.1416; it means "arc of 3.1416 radii." In any particular instance, the context will show to what refers, whether to angle or arc.

It is evident from the second definition of radian measure that, like the trigonometric ratios, the radian measure of an angle is also a ratio of one line to another, namely, the ratio of the subtended circular arc to its radius.

1

NOTE 2. If the radius be taken as unit length, then, by (8) or (5), the number of units of length in the arc is the same as the number of radians in the angle.

EXAMPLES.

8. What is the radian measure of the angle which at the centre of a circle of radius 14 yd. subtends an arc of 8 in.?

in degrees.

Also express the angle

Let denote the radian measure of the angle. Then