28. Shew that a second is approximately equal to '000048 of a radian.

67. The angles

π π П

are the equivalents in radian

4' 3' 6

measure of the angles 45°, 60°, 30° respectively.

Hence the results of Arts. 34 and 35 may be written as

68. When expressed in radian measure the complement

of is

π

2

--

6, and corresponding to the formulæ of Art. 38 we

now have relations of the form

69. By means of Euc. 1. 32, it is easy to find the number of radians in each angle of a regular polygon.

Example. Express in radians the interior angle of a regular polygon which has n sides.

The sum of the exterior angles=4 right angles. [Euc. 1. 32 Cor.] Let be the number of radians in an exterior angle; then

But interior angle=two right angles - exterior angle

13. Find the number of radians in each exterior angle of

(1) a regular octagon,

(2) a regular quindecagon.

14. Find the number of radians in each interior angle of

70. To prove that the radian measure of any angle at the

subtending arc

centre of a circle is expressed by the fraction radius

Let AOC be any angle at the centre of a circle, and AOB a radian; then radian measure of AOC

71. If a be the length of the arc which subtends an angle of radians at the centre of a circle of radius r, we have seen in the preceding article that

The fraction

arc

radius

is usually called the circular measure of the angle at the centre of the circle subtended by the arc.

The circular measure of an angle is therefore equal to its radian measure, each denoting the number of radians contained in the angle. We have preferred to use the term radian measure exclusively, in order to keep prominently in view the unit of measurement, namely the radian.

NOTE. The term circular measure is a survival from the times when Mathematicians spoke of the trigonometrical functions of the arc. [See page 80.]

Example 1. Find the angle subtended by an arc of 7.5 feet at the centre of a circle whose radius is 5 yards.

Let the angle contain ◊ radians; then

Example 2. In running a race at a uniform speed on a circular course, a man in each minute traverses an arc of a circle which subtends 24 radians at the centre of the course. If each lap is 792 yards, how long does he take to run a mile?

Let a yards be the length of the arc traversed in each minute; then from the formula a=r0,

that is, the man runs 360 yds. in each minute.

Example 3. Find the radius of a globe such that the distance measured along its surface between two places on the same meridian 22 whose latitudes differ by 1° 10' may be 1 inch, reckoning that == 7'

Let the adjoining figure represent a section of the globe through the meridian on which the two places P and Q lie. Let O be the centre, and denote the radius by r inches.