199. AB=1263. AC=1359. BC=1468. 200. AB=12.6356. HEIGHTS AND DISTANCES. 201. At 120 feet distance from the foot of a steeple, the angle of elevation of the top was found to be 60° 30'. Required the height. 202. From the top of a rock 326 feet above the sea, the angle of depression of a ship's bottom was found to be 24°. Required the distance of the ship. 203. A wall is surrounded by a ditch; from the edge of this ditch the angle of elevation of a point on the top of the wall is found to be 35°; and at a distance of 100 yards from the ditch the angle of elevation of the same point is found to be 15°. Find the height of the wall; the breadth of the ditch; and the length of the ladder that would just reach from the edge of the ditch to the top of the wall. 204. From the top of a hill I observe two milestones in a straight line before me; and find their angles of depression to be respectively 5° and 15°. Find the height of the hill. 205. Two observers, on the same side of a balloon, and in the same vertical plane with it, a mile apart, respectively find its angles of elevation to be 15° and 65° 30' at the same instant. Find the height of the balloon. 206. A ladder 38 feet long, just reaches to a window 29 ft. 6 in. high on one side of a street; and, on turning the ladder over without moving its foot it reaches a window 28 feet high on the other side. Find the breadth of the street. 207. The top of a maypole being broken off, struck the ground at a distance of 133 feet from the bottom of the pole; and the broken piece was found to measure 294 feet. Find the original height of the pole. 208. The aspect of a wall 18 feet high is due south, and the length of the shadow cast on the north side at noon is 16 feet. Find the sun's altitude, or the angle of elevation of the sun above the horizon. 209. At a distance of 200 yards from the foot of a church tower, the angle of elevation of the top of the tower was 30°, and of the top of the spire on the tower 32°. Find the height of the tower and of the spire. 210. Two men are surveying: when each is at a distance of 200 yards from the flag-staff, the one finds the angle subtended by the position of his companion and the staff to be 30° 15'. Find how far they are apart. 211. In order to ascertain the height of a castle on the top of a cliff, I measured 240 yards directly from it, and at the two ends of this line, found the angles of elevation of the top of the castle to be 29° and 13° 16' respectively; also at the further end of the line, the castle's height subtended an angle of 5° 15'. Find the height. 212. Wishing to know the breadth of a moat surrounding a fortress, I measured a ground line of 30 yards by its side, and at the two ends found the angles subtended by the other end and a corner of the wall close by the other side of the moat to be 32° 19' and 37° 46' respectively, while the angle of elevation at the first end was 58° 32'. Find the breadth of the moat. 213. Two ships, half a mile apart, find that the angles subtended by the other ship and a fort, are respectively 56° 19' and 63° 41'. Find the distance of each ship from the fort. 214. From the top of a house, 42 feet high, I found the angle of elevation of the top of a neighbouring steeple to be 14° 13′, and at the bottom of the house it was 23° 19′. Find the height of the steeple. 215. In walking towards a certain object, I found the angle of elevation of its top to be 2° 19′ 13" at one milestone, and after proceeding to the next milestone, I found the angle of elevation to be 3° 28' 49". How much further should I have to walk before I reached it? 216. Wishing to ascertain the height of a house standing on the summit of a hill, I descended the hill for 40 feet, and then found the height subtended an angle of 34° 18′ 19′′. On descending still further a distance of 60 feet, I found the same angle to be 19° 14' 52". Find the height of the house. 217. Wanting to know the height of a castle on a rock, I measured a base line of 100 yards, and at one extremity found the angle of elevation of the castle's top to be 45° 15′, and the angle subtended by the castle's height to be 34° 30'; also the angle subtended by the top of the castle and the other extremity of the base line to be 73° 14'. At the other extremity the angle between the first extremity and the top of the castle was 73° 18'. Find the height of the castle. 218. Having measured a base line of 400 yards, whose upper end was 24 feet higher than the lower one, in the same vertical plane with the top of a hill, I found the angles of elevation of the top of the hill from the lower and upper ends of the base line to be 5° 14' and 3° 17' respectively. Find the height of the hill. 219. In order to measure the distance between two inaccessible objects C and D, I measured a base line AB of 500 yards, and at its extremities determined the following angles: CAB=94° 13′, DAB=62° 20′, DBA=84° 58', and CBA=41° 16'. Find the distance between C and D. 220. Wanting to know my distance from an object P on the other side of a river, and having no instrument for observing angles, I measured a base line AB of 500 yards, and from A and B measured directly in a line away from P distances of 175 yards to C and D; I then found my distances from B and A to be respectively, CB=500 and AB=650 yards. Find PA and PB. 221. A lighthouse was observed from a ship to bear N. 34° E., and after the ship had sailed due south for 3 miles, the same lighthouse bore N. 23° E. Find the distance of the lighthouse from each position of the ship. 222. Two objects, A and B, were observed from a ship to be at the same instant in a line with a bearing N. 15° E. The ship then sailed N.W. for 5 miles, when it was observed that A bore E., and B bore N.E. Find the distance between A and B. 223. A privateer is lying 10 miles S.W. of a harbour, and observes a merchantman leave it in the direction of S. 80° E., at the rate of 9 miles an hour. In what direction, and at what rate, must the privateer sail in order to come up with the merchantman in I1⁄2 hours? 224. From the top of the peak of Teneriffe, the dip of the horizon is found to be 1° 58' 10". If the radius of the earth be 4000 miles, what is the height of the mountain? 225. What is the dip of the horizon from the top of a mountain 13 miles high, the radius of the earth being 4000 miles? 226. From the top of a mountain 1 miles high, the dip of the horizon was found to be 1° 34' 30". What is the diameter of the earth? 227. In a town are three remarkable objects, A, B and C, known to be distant from each other as follows: AB=426.75, AC=610, and BC=538.5. From my position S I observe that B lies beyond the line AC, and within the angle ASC: and I find the angle ASC=23° 9', and ASB=14° 16'. Find the distance of S from A, B and C respectively. 228. Having removed to the other side of the town, so that B lies on the side of AC next to S, and still within the angle ASC; I observe the angle ASC to be 15° 14′ and 14° 15'. Find SA, SB, and SC. 229. Having again moved so as to have A and C in a line with S (A being the nearer), I find the angle ASB to be 18° 17'. Find SA, SB, and SC. 230. Three points of land, A, B and C, are at known distances from each other, namely AB=63, AC=44, and BC=76. At a boat in the piece of water between them the angles subtended by AB and BC are observed to be 89° 15′ and 130° 45′ respectively. Find the distance of the boat from A, B and C. 231. Being on a river, and observing a column on the banks, I find the angle of elevation of its top to be 30°, and the angle subtended by its top and a small island down the river to be 47° 25'. After sailing past the column to this island, a distance of 450 yards, I find the angle subtended by the top and my former position to be 18° 30'. Find the height of the column. 232. On the opposite bank of a river to that on which I stood, is a tower 216 feet high. With a sextant I ascertained the angle subtended at my eye by the height of the tower to be 47° 56'. Find my distance from the foot of the tower, supposing my eye to be 5 feet above the level of the tower's foot. EXPANSIONS, SERIES, ETC. 233. Expand (cos ) in terms of the cosines of and its multiples. 234. Expand (cos ) 27 in terms of the cosines of multiples of 0. Thence find (cos 0), (cos 0)4, (cos 0), (cos 0), (cos 0), (cos 0) 12. 235. Expand (cos )2n+1 in terms of the cosines of multiples of 0. Thence find (cos )3, (cos 0)5, (cos 0)7, (cos 0)?, (cos 0)11. 236. Expand (sin 0)4m in terms of the cosines of multiples of 0. Thence find (sin 0)4, (sin 0), (sin 0)12. 237. Expand (sin 0)4m+1 in terms of the sines of multiples of 0. Thence find (sin 0)5, (sin ). 238. Expand (sin 0)4m+2 in terms of the cosines of multiples of 0. Thence find (sin 0), (sin 0), (sin 0)1o. 239. Expand (sin 0)4m+3 in terms of the sines of multiples of 0. Thence find (sin 0)3, (sin 0)7, (sin )". 240. Find sin 40, and sin 90 in terms of the powers of sin and cos 0. 241. Find cos 5 0, cos 60 and cos 70 in terms of the powers of sin and cos 0. 242. Find tan 30, and tan 80 in terms of the powers of tan 0. From the exponential expressions for sin and cos 0, obtain the following formula: .... 249. Sum the series I+cosx+ cos 2x + cos 3x+ to n terms. 250. Sum the series 1+x cos 0+x2 cos 20+x3 cos 30+.... to n terms. 251. Sum the series sina + sin 2a + sin 3a+ .... to n terms. 252. Sum the series cos 0+ cos 30+cos 50+.... to infinity. .... 23 254. Sum the series tan0+2 tan 20+ 22tan 40+ to n terms. 255. Sum the series coseca + cosec 2a + cosec4a+cosec8a+ .... to n terms. 256. Sum the series 1 + cose. cos + cos20. cos 24+ cos3 0.cos 34 +.... to infinity. 257. Sum the series cose + cos (+7) + cos (0 +24) + cos (+34) +.... to n + 1 terms. 259. Sum the series cos20+ cos(0+a) + cos2 (0+2α) +cos2 (0+3a) +.... to n terms. 260. Sum the series sin 45°. sin0+ sin2 45°. sin 20 + sin345°. sin 30+ to n terms. - sine)+(cos 261. Sum the series (cos0+ I sin0) + (cos + √ I sin 0)2 +(cos +I sin 0)3+.... (1) to n terms, and (2) to infinity. I 5 2 262. Sum the series (3a—3 ̃3) — ''(3*—3 ̄1) + (3*—3 ̄3) 263. Prove that tan no= whether n be even or odd. 5 sin0+ sin 30+ sin 50+.... to n terms cose+cos 30+ cos 50+.... to n terms 264. If sin an sin (a+x); express x in a series of sines of a and its multiples. 265. If tan0= m cos 1+m sin; express in a series of sines and cosines of multiples of p. 266. If a and b be two sides of a triangle, opposite to the angles A and B, then loge=(cos 2B―cos 2A) + 1⁄2 (cos 4B — cos 4A) |