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which will correspond to the same position of OB as does. The continuation of the series will be

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showing that the positions will be continually repeated in regular order.

EXERCISES.

1. If AOB = 30°, or 30° + C, or 30°+ 2C, etc., at what angles will AOB fall?

2. How many degrees between the minute-lines on a clock-face? 3. At what angles with XII. are the hour and minute hands of a clock together?

4. If the hour-hand is so displaced that when the minute-hand is at XII. the hour-hand is 2m past XII., at what angles will the hands be together?

5. What values may two thirds of the general measure of an angle of 105° have?

6. If an angle is in the third quadrant, what are the limits between which its bisectors must fall?

7. Between what three sets of limits must a be contained in order that 3a may fall in the fourth quadrant?

8. Show that while one sixth of the general measure of an angle has six different values, two sixths has only three values, and three sixths only two values. Show that this diminution arises from several values falling together when multiplied by 2 or 3. As an example, take the case when a = 48°; fa = 8°, 68°, etc.

14. Natural Measure of Angles. The division of the circumference into 360° is entirely arbitrary, and any other angle than the degree may be taken as the unit.

In purely mathematical investigations, where

no division into degrees is required, the length

of the radius is taken as the unit of measure.

This unit is called the radian.

B

The radian is therefore the angle subtended An angle of one radian

by an arc whose length is equal to the radius.

in which arc XB radius OX.

To find the relation of this unit of angle to the degree,, minute, and second, we note that the ratio of the entire circumference to the diameter is 3.141 592 65, etc. (Geom., Book

VI., §5.)

Hence its ratio to the radius is double this number, or 6.283 185 3, etc. Since the circumference measures 360°, the unit

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Hence, when we take the radian as the unit,

represents an angle of 90°;

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Considering the radius of the circle as unity, what is the length of circular arcs subtending the following angles?

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NOTE. In these exercises the angle is first to be reduced to a common denomination of measure, either degrees, minutes, or seconds. For instance, 28° 17′ 15′′.6 = 101 835′′.6 = 1697'.26 - 28°.287 67.

If the radius is 100 metres, how many degrees and minutes will arcs of the following lengths subtend?

5. 100 metres. Ans. 57° 17′ 44′′.8.

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With what radius will

8. Arc of 32 metres' length subtend an angle of 32°?

Ans. 57.296.

9. Arc of 32 metres' length subtend an angle of 32'?

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They are to be

12. Two railways met at right angles at O. connected by a quadrant PQ, of which the inner rail shall be 600 metres in length. What is the common distance OP and OQ of the switches from the point in which the two inner rails would meet? Ans. 381.97. 13. In the preceding case, if the rails are 5 feet apart, how much longer will the outer rail of the curve be than the inner one?

Ans. 7.854 feet.

14. Show that if three circles, equal or unequal, mutually touch each other externally in the points A, B, and C, the sum of the three included arcs AB+BC + CA, expressed in angular measure, is equal to a certain constant. What is this constant?

15. If the two lesser circles, still touching each other, touch the greater one internally, show that the sum of their arcs minus the arc of the greater circle, expressed in angular measure, is equal to the same constant as that of the preceding problem.

16. The earth's equatorial diameter being 12,756 kilometres, what is the length of one degree of the equator in kilometres and in miles, assuming 1 metre = 39.37 inches.

17. Explain why a degree of latitude is greater at the poles than at the equator, although the radius of the earth is less.

Remark. At this stage of his progress, if not sooner, the student should be familiarized with the use of the logarithmic and trigonometric tables, and should employ them in all computations in which they are applicable

CHAPTER II.

THE TRIGONOMETRIC FUNCTIONS.

The Sine, Tangent, and Secant.

15. To investigate the numerical relations between the sides and angles of geometric figures, certain functions of angles are employed in trigonometry. These functions are defined in the following way:

Let OX be that side of the angle from which we measure, the length OX being taken as the radius of a circle. Also, suppose OB, the side to which we measure;

M, the point in which OB intersects the circle;
XQ, the line tangent to the circle at X;

N, the point at which

this tangent meets the side OB;

MP, the perpendicular from M upon OX. Then taking the radius OX as unity, and expressing other lengths in terms of this unit:

I. The length MP

is called the sine of the

angle XOB.

II. The length NX

is called the tangent

of the angle XOB.

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III. The length ON is called the secant of the angle XOB.

The absolute lengths of the lines representing the sine, tangent, and secant, considered as lines, will vary with the radius of the circle. This is avoided by taking for the sine, tangent, and secant, not the lines which represent them, but the ratios of the lines to the radius of the circle, which ratios will be pure numbers.

We have now to prove that these numbers are the same for the same angle whatever be the radius.

Let X'ON' be the angle.

From the vertex O draw the two

arcs XM and X'M' with any two radii OX and OX.

N

M

N'

P X P' X

Erect the respective sines and tangents PM, XN, P'M', X'N'. Then because the triangles OPM, 0XN, OP'M', and OX'N' have the angle at O common, and the respective angles at P, X, P', and X' all right angles, and therefore equal, these triangles are equiangular and similar.

Comparing the sides about the equal angles we have the ratio

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The equations (a) now show that the sine, tangent, and secant of the angle will be represented by the same numbers whether we measure them in the inner or outer circle. Therefore:

To each angle of a definite magnitude corresponds One definite number, called the sine of the angle; Another definite number, called the tangent of the angle; Another definite number, called the secant of the angle.

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