77. To find the Trigonometrical Ratios for an angle of 90°. We have already proved in the preceding Chapter that = If a=0°, B = 30°, y = 45°, S= 60°, 90°, find the numerical values of the following expressions: 78. We are now able to give some simple examples of the practical use of Trigonometry in the measurement of heights and distances. 79. The values of the sines, cosines, tangents and the other ratios have been calculated for all angles succeeding each other at intervals of 1', and the results registered in Tables. Instruments have been invented for determining: 1. The angle which the line joining two distant objects subtends at the eye of the observer. 2. The angle which a line joining the eye of the observer and a distant object makes with the horizontal plane. If the object be above the observer the angle is called the Angle of Elevation. If the object be below the observer the angle is called the Angle of Depression. 80. To find the height of an object standing on a horizontal plane, the base of the object being accessible. Let PQ be a vertical column. From the base P measure a horizontal line AP. Then observe the angle of elevation QAP. We can then determine the height of the column, for QP-AP. tan QAP. Let RS be the horizontal line joining two objects on the opposite banks. From O, a point in a vertical line with R, observe the angle of depression OSR. Then if OR be measured we can determine the length of RS, for 82. To find the height of a flag-staff on the top of a tower. R Let RQ be the flag-staff. From P the base of the tower measure a horizontal line AP. Observe the angles RAP and QAP. Then we can find the length of RQ, for RQ = RP-QP =AP. tan RAP-AP, tan QAP. 83. To find the altitude of the Sun. The altitude of the Sun is measured by the angle between a horizontal line and a line passing through the centre of the Sun, S. T. 4 If AB be a stick standing at right angles to the horizontal plane QR, and QB the shadow of the stick on the horizontal plane, a line joining QA will pass through the centre of the Sun. Then if we measure AB and QB we shall know the altitude of the Sun, for (1) At a point 200 feet from a tower and on a level with its base the angle of elevation of its summit is found to be 60°: what is the height of the tower? (2) What is the height of a tower whose top appears at an elevation of 30° to an observer 140 feet from the foot of the tower on a horizontal plane, his eye being 5 feet from the ground? (3) Determine the altitude of the Sun when the length of a vertical stick is to the length of the shadow of the stick as √3:1. (4) At 300 feet measured horizontally from the foot of a steeple the angle of elevation of the top is found to be 30o. What is the height of the steeple? (5) From the top of a rock 245 feet above the sea the angle of depression of a ship's hull is found to be 30°. How far is the ship distant? (6) From the top of a hill there are observed two consecutive milestones, on a horizontal road, running from the base. The angles of depression are found to be 45° and 30°. Find the height , of the hill. (7) A flag-staff stands on a tower. I measure from the bottom of the tower a distance of 100 feet. I then find that the top of the flag-staff subtends an angle of 45o and the top of the tower an angle of 30° at my place of observation. What is the height of the flag-staff? (8) From the summit of a tower whose height is 108 feet, the angles of depression of the top and bottom of a vertical column, standing on a level with the base of the tower, are found to be 30° and 60° respectively. Find the height of the column. (9) A person observes the elevation of a tower to be 60o, and on receding from it 100 yards further he finds the elevation to be 30°. Required the height of the tower. (10) A stick 10 feet in length is placed vertically in the ground, and the length of its shadow is 25 feet. Find the altitude. of the Sun, having given tan 25° = 4. (11) A spire stands on a tower in the form of a cube whose edge is 35 feet. From a point 23 feet above the level of the base of the tower, and 20 yards distant from the tower, the elevation of the top of the spire is found to be 56°. 34. Find the height of the top of the spire, having given (12) The length of a kite string is 250 yards and the angle of elevation of the kite is 30o. Find the height of the kite. (13) The height of a house-top is 60 feet. A rope is stretched from it and is inclined at an angle of 40o. 30′ to the ground. Find the length of the rope, having given sin 40°. 30' '65. (14) A tower on the bank of a river is 120 feet high, and the angle of elevation of the top of the tower from the opposite bank is 20°; find the breadth of the river, having given tan 20o = •35. (15) The altitude of the Sun is 36°. 30': what is the length of the shadow of a man 6 feet high, if tan 36o. 30′ = •745 ? |