. Elements of plane and spherical trigonometry . point of the earths surface is takenand found to be 2° 13 27; it is required thence to determine the dia-meter of the earth, supposing it to be a perfect sphere. Let O be the centre of the earth, BA the mountain, ^ AC the visual ray or line touching the earths surface b^^ ™ in C. Draw the tangent BD, and join OD, OC ; then j^ //^the angle of depression EAC being given, we have / 1/ ys also the angle BAD, the complement of it, equal to / r ] 87° 46 33. Also since the tangents BD, CD, are I ^ /equal, (Geom. p. 106,) we have the angle BOD = V / DOC
. Elements of plane and spherical trigonometry . point of the earths surface is takenand found to be 2° 13 27; it is required thence to determine the dia-meter of the earth, supposing it to be a perfect sphere. Let O be the centre of the earth, BA the mountain, ^ AC the visual ray or line touching the earths surface b^^ ™ in C. Draw the tangent BD, and join OD, OC ; then j^ //^the angle of depression EAC being given, we have / 1/ ys also the angle BAD, the complement of it, equal to / r ] 87° 46 33. Also since the tangents BD, CD, are I ^ /equal, (Geom. p. 106,) we have the angle BOD = V / DOC = i comp. A = 1° 6 49J, and, therefore, \ ^ BDO = 88° 53 161. Now in the right-angled triangle ABD we have BD = AB tan. A; and in the right-angled triangle OBD, OB = BD tan. BDO; hence by substitution, OB = AB tan. A tan. BDO; the computation is, therefore, as follows: AB = 3 . 0 4771213 tan. A 87° 46 33 . 11-4107381 tan. BDO 88° 53 16^ . 117119309 OB 3979-15 . 3-5997903; hence the diameter is 7958-3 * 30 PLANE TRIGONOMETRTo. PROBLEM VI. Given the distances between three objects A, B, C, and the anglessubtended by these distances at a point D in the same plane with them;to determine the distance of D from each object. Let a circle be described about the triangle ADB, and join AE, EB,then will the angles ABE, BAE, be respectively equalto the given angles ADE, BDE, ( 52); thus allthe angles of the triangle AEB are known, as also theside AB; we may find, therefore, the remaining sidesAE, EB. Again, the sides of the triangle ABC beingknown, we may find the angle BAG ; hence the angleCAE becomes known, so that in the triangle CAE weshall have the two sides AE, AC, and the includedangle given, from which we may find the angle AECin fig. 1, or the angle ACE in fig. 2, and thence itssupplement AED or ACD; this with the given sideAE and angle ADE, in the first figure, or with thegiven side AC and angle ADC in the second, willenable us to find AD, one of the requir
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