Example 1.—Having measured a distance of 200 feet in a direct horizontal line from the bottom of a steeple, the angle of elevation of its top, taken at that distance, was found to be 47° 30′, from hence it is required to find the height of the steeple?
By deducting 47° 30′ from 90°, the angle opposite the given side will be found (42° 30′).
Then by Case 1. Trigonometry:—
| As | sine ∠ 42° 30′ | : 200 | :: sine ∠ 47° 30′: |
| Or | ·67556 | : 200 | :: ·73723 : 208·2, &c., height required. |
By construction—
The triangle is constructed by making the side from a scale of equal parts, and laying down the angles from the protractor. Then by measuring the unknown parts by the same scale, the solution will be obtained.
Example 2.—Being on the side of a river, and requiring the distance to a house on the other side, 200 yards were measured in a straight line by the side of the river, and at each end of this base line the angles with the house were 68° 2′, and 73° 15′—required the distance from each end of the base line to the house?
The sum of the given angles (68° 2′ + 73° 15′) subtracted from 180° will give the third angle (38° 43′).
Then by Case 1. Trigonometry:—
| As sine ∠ 38° 43′ | : 200 | :: sine ∠ 68° 2′ |
| ·62544 | : 200 | :: ·92739 : 296·5 first distance required. |
| As sine ∠ 38° 43′ | : 200 | :: sine ∠ 73° 15′ |
| ·62544 | : 200 | :: ·95753 : 306·1 second distance required. |