(7) sin A-sin B+sin C=4 sin A cos B sin C.

(9) sin 2A+sin 2B+sin 2C=4 sin A sin B sin C.

(10) (a+b) cos C + (a+c) cos B + (b + c) cos A=

(13) a sin (B-C)+b sin (C-A)+c sin (A−B)=0.

(16) Areas. (s—a). (s—b). (8-0).

(17) (a+c) sin 1⁄2 B=b. cos ↓ (A–C).

(18) sin A+ sin B+sin2C-2 cos A cos B cos C=2.

97. The sum of the squares of the cosines of the angles of a triangle is equal to unity; show that the triangle is rightangled.

98. Having given the perimeter 2s, and the angles A, B, C, of a plane triangle, find its sides.

CXVII.

HEIGHTS AND DISTANCES

1. The angle of elevation of the top of a steeple is 60° from a point on the ground. That of the top of the tower on which the steeple rests is 45° from the same point. What proportion does the height of the steeple bear to that of the tower?

2. The angle of elevation of the top of a steeple is 45° from a point in the same horizontal plane as its base, and is 30° from a point 30 ft. directly above the former point. Find the height and distance of the steeple.

3. An object C being inaccessible from B, a line BA, 408 yards in length, is measured, and the angles CBA, CAB, observed to be 22° 37′ and 58° 7' respectively. Find BC, having given L sin 80° 44′=9.994295, L sin 58° 7′ =9.928972, log 4.08=610660, and log 3.51=545337.

4. A pole is fixed on the top of a mound, and the angles

of elevation of the bottom and top of the pole are 30° and 60° respectively. Prove that the height of the pole=twice the height of the mound.

5. 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 foot-path which goes the shortest way to the canal, what is the distance by it?

6. What is the angle of depression of an object? From the top of a hill the angles of depression of two consecutive milestones on a straight, level road, were found to be 12° 13' and 2° 45'. Find the height of the hill.

7. A balloon ascends uniformly in a vertical line; an observer takes up a position in the horizontal plane at a distance of 350 yards from the point where the balloon left the earth; after a time he observes the angle of elevation of the balloon to be 55° 17', and after an interval of three minutes he observes the angle of elevation to be 60° 18'. Find the rate of the ascent of the balloon.

8. A lighthouse was observed by a ship at sea to bear SE.; after the ship had sailed NE. for 12 miles, the lighthouse was observed to bear 15° E. of S. Find the distance of the lighthouse from each position of the ship.

9. An object 10 ft. high is placed on the top of a tower, and subtends an angle of 6o at a place which is in the same horizontal plane as the foot of the tower, and is 50 ft. distant from it. Determine the height of the tower.

10. From a boat on a river, the angle of elevation of the top of a column on the bank was observed to be 32° 15', and the angle subtended by the top of the column and a boat down the river was 48° 12'; after sailing past the column towards the barge for a distance of 480 yds., the observer in the boat found that the angle subtended by the top of the column and the first position of the boat was 20° 20′. Find the height of the column.

11. A person, wishing to determine the length of an inaccessible wall, places himself due S. of one end and then due W. of the other, at such distances that the angle which the length of the wall subtends at each station is 30°. Find the length of the wall, the distance between the stations being 120 yds.

12. Four points A, B, C, D are in the same horizontal plane; if the distance CD be known, show how to determine AB by means of angles measured at A and B.

13. An object 12 ft. high, standing on the top of a tower, subtends an angle of 1° 54' 10", at a station which is 250 ft. from the base of the tower. Find the height of the tower.

14. From a ship sailing due SE. at the rate of seven 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.

15. A man ascends a hill by a path which is the shortest distance between the base and the summit; the inclination of the path to the horizon is at first 20°, but afterwards suddenly increases to 40°; on arriving at the summit he observes the angle of depression of the point from which he started to be 60°. Supposing that he walked 800 yds. before the change of inclination of his path, determine the height of the hill.

16. B and C are two stations 3 miles apart; travellers start from B and C towards a third station, A: the traveller from B, walking 4 miles an hour, reaches A in hour; the traveller from C, at the rate of 5 miles an hour, reaches A inhour. Find either angle of the triangle ABC.

17. An observer, whose eye is 5 ft. above the ground, observes a vertical object 100 ft. high standing on the same horizontal plane with him, and finds the angular distance between its highest and lowest points to be 30°. What is the distance of the object from him?

18. An observer, standing on a horizontal plane, observes

the angle of elevation of the top of a mountain to be 63° 25′; walking 1 mile towards the mountain, on an ascent that makes an angle of 30° with the horizon, he finds the eleva tion of the mountain top to be 74° 25'. Find the height of the mountain above the horizontal plane.

19. AB, AC, are two railroads inclined at an angle of 50° 20′; a locomotive engine starts from A along AB at the rate of 30 miles an hour; after an interval of 1 hour, another locomotive engine starts from A along AC, at the rate of 45 miles an hour. Find the distance between the engines 3 hours after the first started.

20. Wishing to know the distance of an inaccessible object P, and having no instrument for measuring angles, I measured a horizontal base AB, 400 ft., and from A to B, in a line directly away from P, I measured AC, 300 ft., BD, 450 ft.; I also measured BC, 500 ft., and AD, 750 ft. Find PA or PB.

21. The elevation of a tower standing on a horizontal plane is observed; on advancing 240 ft. nearer to it, the angle of elevation is found to be the complement of the first observed angle, and on advancing 100 ft. nearer still to the tower, its angle of elevation is found to be double the first. Find the height of the tower.

If the given distances be a and b, between what limits must the first angle of elevation lie, that the angular data, as above, may be possible?

22. The angle of elevation of a tower 400 ft. high, when due N. of the observer, was 55° 14'; what will the elevation be when the observer has walked 720 ft. due E. on the horizontal plane?

23. A railroad AB runs N., and a railroad AC NE. from A. A train starts from A along AB with a velocity of 25 miles an hour; after half an hour a train starts from A along AC with a velocity of 30 miles an hour. Find the distance of the trains from each other three hours after the first train has started.