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spectively; prove that the height of the pole is twice the height of the mound.

7. Two straight roads, which cross one another, meet a canal at angles of 60° and 30° respectively. If it be 3 miles by the longer of the two roads, from the crossing to the canal; how far is it by the shorter? If there be a footpath which goes the shortest way to the canal, what is the distance by it?

8. A ladder 30 feet long reaches a window 24 feet high on one side of a street; turning over the top of the ladder to the other side of the street without moving its foot, it exactly reaches a window when its new position is at right angles to its former one. Required the width of the street and the height of the latter window.

9. A cruiser from a point N.E. of a harbour observes a blockaderunner in a direction 15° E. of S., running straight for the harbour with a N.W. course; which it pursues, and overtakes at the mouth of the harbour. Compare the rates of sailing.

10. From a ship sailing due S.E. at the rate of 7 miles per hour, a lighthouse is observed to bear N. 30° E., and after two hours its bearing is due N.; find the distance of the ship from the lighthouse at each observation.

11. A privateer descries a barque 12 miles off to the N.E. sailing with a N.W. course, and pursues and overtakes it running due N. for 8 miles, and W. 15° from N. for the remainder of the How far did each vessel run in the pursuit ?

race.

12. From the top of a ship's mast, 125 feet above its hull, the angle of depression of the hull of another ship is found to be 6° 50′ 34". Find the distance between the ships.

Given tan 6° 50′ 34′′ = 12.

13. A kite has attained an altitude of 36°, and 200 yards of string are out; required the perpendicular height of the kite, supposing the string to be perfectly stretched.

Given sin 36° = •5878.

14. A castle wall, 20 feet high, is surrounded by a ditch; from the edge of which the altitude of the top of the wall is 60°. Find the width of the ditch, and also how far an observer must walk directly backward in order that the altitude of the wall may be 30°.

Given √3 = 1·732.

15. Starting from a point at which I perceive a steeple due E., and a tower due W., I travel for 5 miles and then observe the steeple to be S.E., and the tower to be 15° from S. How far is it from tower to steeple ? Given cot 75° =

16. From the top of

26 nearly.

a hill the angles of depression of two con

secutive milestones on a straight, level road, were found to be 12° 13′ and 2° 45': find the height of the hill.

Given tan 2° 45' =

048, tan 12° 13'

= .217.

17. From the edge of one bank of a river a person ascends 100 yards up a slope of 1 in 4, and observes the angle of depression of an object on the opposite bank close to the edge of the river to be 14. Find the breadth of the river.

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18. Solve the right-angled triangle ABC, given a + b and A. 19. Also solve the triangle, given a-b and A.

20. Find a, given b-c and A.

21. Solve, given a+b+c and A.

CHAPTER VI.

THE TRIGONOMETRICAL ANGLE AND ITS FUNCTIONS IN THE FOUR QUADRANTS.

35. The Trigonometrical and Geometrical angle not identical.

Let OP, starting from the position OA, revolve about O, and assume successively the positions, OP, OP, OP3, ... OPm; where OPm is in the same straight line with OA. P Draw OB at right angles to OA. The angular spaces revolved through by OP are measured successively by the angles

m

P

P

A

m'

AOP, AOP,, &c., until it arrives at the position OP We know it has then revolved through two right angles AOB and BOPm; and we should infer, from what we have just seen, that AOPm is an angle equal to two right angles. But AOP is not an angle at all by the geometrical definition (Euc. I. def. 9). In our present subject, however, the definition of an angle which is made to suffice for the purposes of Geometry would lead to much inconvenience; and in consequence, it is "extended," as will now be seen.

36. Extension of the definition of an angle.

In Trigonometry it is convenient to imagine all our angles traced out, as above, by the turning of a line

about a point from one position to another. Assuming that Geometry furnishes us with our conception of "angular space," we may then define, as follows, an

Angle in Trigonometry. OA being the first position of OP, the angle AOP is the angular space revolved through by OP in coming to its present position.

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OP may of course revolve in either of two directions, upwards from OA, or downwards, and form angles of all magnitudes, either as in Fig. 1, or as in Fig. 2.

m

Hence, in the present subject, AOP (figure to Art. 35) is an angle; and is measured by two right angles, or 180°, or π. And continuing the revolution before spoken of until OP again coincides with OA, we meet further with angles of all magnitudes between two and four right angles-between 180° and 360°, or π and 2π. More than this, still continuing the revolution about 0, we obtain, consistently with our definition, angles greater than four right angles; and it is evident that there is no limit to the magnitude of the trigonometrical angle.

The revolving line is frequently said to generate the angle, and is itself spoken of as the generating line.

37. It may be interesting to note that Euclid's defi

nition of an angle is not always adhered to even ín Euclid's Elements. Into the demonstration of Prop. 33, Book VI. two angles are introduced; each of which is any multiple of another given angle.

If the geometrical angle POP (figure to Art. 35.) be 120°, the trigonometrical angle AOP, indicated by the dotted arc of a circle

= LAOPm+120°

180°+120° = 300°.

If the generating line be supposed to have made one complete revolution before taking up its present position, the trigonometrical angle generated will then

= 360°+300° = 660°.

After two revolutions it would

= 2 × 360° +300°

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1020°.

As this mode of generating angles is not restricted, the line OA and OP or any other two lines under similar conditions, may bound an infinite number of trigonometrical angles.

Speaking generally, if A be any angle less than 360°, the expression for all angles bounded by the same two straight lines as A is

n. 360° +A;

or, using circular measure, the general expression for all angles bounded by the same straight lines as a is

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39. The Trigonometrical Ratios of Angles generally.

It was found convenient in a previous chapter to treat

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