*71.

To prove that the measure of an angle at the centre of a circle in radians (i.e. in Circular Measure) is the ratio of the arc on which it stands to the radius of the circle.

L

radian

radius

R

Since angles at the centre of a circle are to one another as the arcs on which they stand (Euc. VI. 33),

Therefore any angle ROP at the centre of a circle is to the radian as its arc RP is to the arc of the radian.

But the arc of the radian is equal to the radius,

Therefore

any angle ROP
is equal to
the radian

And therefore any angle ROP is equal to

radian).

That is, the measure a of an angle in radians is the ratio

Example. Find the number of grades in the angle subtended by an arc 46 ft. 9 in. long, at the centre of a circle whose radius is 25 feet.

The angle stands on an arc of 463 ft. and the radian, at the centre of the same circle, stands on an arc of 25 feet.

* EXAMPLES. X.

(In the answers 22 is used for π.)

(1) Find the number of radians in an angle at the centre of a circle of radius 25 feet, which stands on an arc of 371⁄2 feet.

(2) Find the number of degrees in an angle at the centre of a circle of radius 10 feet, which stands on an arc of 5 feet.

(3) Find the number of right angles in the angle at the centre of a circle of radius 3 inches, which stands on an arc of 2 feet.

(4) Find the number of French minutes in the angle at the centre of a circle of radius 8 ft. 4 inches, which stands on an arc of 1 inch.

(5) Find the length of the arc subtending an angle of 41 radians at the centre of a circle whose radius is 25 feet.

(6) Find the length of an arc of eighty degrees on a circle of 4 feet radius.

(7) Find the length of an arc of sixty grades on a circle of ten feet radius.

(8) The angle subtended by the diameter of the Sun at the eye of an observer is 32'; find approximately the diameter of the Sun if its distance from the observer be 90,000,000 miles.

(9) A railway train is travelling on a curve of half a mile radius at the rate of 20 miles an hour; through what angle has it turned in 10 seconds?

(10) A railway train is travelling on a curve of two-thirds of a mile radius, at the rate of 60 miles an hour; through what angle has it turned in a quarter of a minute?

(11) Find approximately the number of English seconds con

tained in the angle which subtends an arc one mile in length at the centre of a circle whose radius is 4000 miles.

(12) If the radius of a circle be 4000 miles, find the length of an arc which subtends an angle of 1" at the centre of the circle.

(13) If in a circle whose radius is 12 ft. 6 in. an arc whose length is 6545 of a foot subtends an angle of 3 degrees, what is the ratio of the diameter of a circle to its circumference?

(14) If an arc L-309 feet long subtend an angle of 71 degrees at the centre of a circle whose radius is 10 feet, find the ratio of the circumference of a circle to its diameter.

(15) On a circle 80 feet in radius it was found that an angle of 22° 30′ at the centre was subtended by an arc 31 ft. 5 in. in length; hence calculate to four decimal places the numerical value of the ratio of the circumference of a circle to its diameter.

(16) If the diameter of the moon subtend an angle of 30′, at the eye of an observer, and the diameter of the sun an angle of 32', and if the distance of the sun be 375 times the distance of the moon, find the ratio of the diameter of the sun to that of the

moon.

(17) Find the number of radians in (i.e. the circular measure of) 10" correct to 3 significant figures. (Use § for π.)

(18) Find the radius of a globe such that the distance measured upon its surface between two places in the same meridian, whose latitudes differ by 1o 10', may be one inch.

(19) Two circles touch the base of an isosceles triangle at its middle point, one having its centre at, and the other passing through the vertex. If the arc of the greater circle included within the triangle be equal to the arc of the lesser circle without the triangle, find the vertical angle of the triangle.

(20) By the construction in Euc. I. 1, prove that the unit of circular measure is less than 60o.

(21) On the 31st December the Sun subtends an angle of 32′ 36′′, and on 1st July an angle of 31′ 32′′; find the ratio of the distances of the Sun from the observer on those two days.

(22) Show that the measure of the angle at the centre of a k.a

circle of radius r, which stands on an arc a, is depends solely on the unit of angle employed.

Find / when the unit is (i) a radian, (ii) a degree.

where k

*72. Questions concerning angles expressed in different systems of measurement are easily solved by expressing each angle in right angles.

Example 1. The sum of the measure of an angle in degrees and twice its measure in radians is 234, find its measure in degrees (π=22).

Let the angle contain x right angles.

Then the measure of the angle in degrees=90x,

The angle is of a right angle, that is 2210, nearly.

Example 2. The three angles of a triangle are in arithmetical progression, and the measure of the least in grades is to that of the greatest in circular measure as 120: π. Express each angle in degrees.

Let the angles contain x-y, x, x+y right angles respectively; they are then in A.P.

Their sum is 3x right angles; but since they are the angles of a triangle, their sum is 2 right angles;

.. 3x=2,
..x=3.

Again, the least angle contains (x-y) x 100 grades, and the

..x-y=z-}=1, x=3, x+y=} + } = {.

Thus the angles contain,, and right angles respectively;

therefore the angles are 45°, 60°o, 75o.

**EXAMPLES. XI.

(In the following examples the answers will be given in terms of π.)

(1) The sum of the degrees and of the grades in a certain angle is 38; find its circular measure.

(2) The difference of two angles is 20%, and their sum is 48°; find them.