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88. Examples to the Cases of Plane Triangles.

1. In a right-angled triangle given the hypothenuse 185, and one of the acute angles 32° 40', to find the other parts of the triangle.

2. Given one of the sides 64, and the opposite angle 23° 15'. 3. Given the hypothenuse 324, and one of the sides 265. 4. Given the two sides 78 and 59.

5. In an oblique-angled triangle given one of the angles 69° 14', another 46° 27', and the side between them 248, to find the other parts of the triangle.

6. Given one of the angles 28°, another 114°, and the side between them 57.

7. Given one of the sides 215, another 169, and the angle opposite to the former 74°.

8. Given one of the sides 329, another 248, and the angle opposite to the latter 26°.

9. Given one of the sides 79, another 67, and the included angle 85° 16'.

10. Given one of the sides 241, another 175, and the included angle 103°.

11. Given one of the sides 126, another 148, and the third 96. 12. Given the three sides 119, 196, 175.

89. Mensuration of Heights and Distances.

The mensuration of heights and distances depends upon the use of certain instruments for taking angles, and the rules of plane trigonometry already delivered. By the joint application of these, the heights and distances of terrestrial objects may be determined, and the distances and magnitudes of the heavenly bodies may be discovered.

Horizontal and vertical angles are taken with a theodolite furnished with one or two telescopes and a vertical arc. If the circles of the theodolite be about 31 inches radius, the observed angles may be read off to half a minute of a degree.

If the angles to be taken be oblique to the horizon, they must be taken with a sextant, or Hadley's quadrant, which must be held in such a position that its plane may pass through both objects and the eye of the observer. Elevations are taken with the quadrant by reflecting the object from an artificial horizon.

Angles of elevation, or of depression, are commonly taken with a quadrant having its arch divided into degrees and mi

F

nutes, and a plummet suspended from the centre, and two sights fixed perpendicularly upon one of its radii.

Note. The use of instruments in taking angles cannot be easily taught by written directions, and is best learned from some person who is acquainted with their use.

Practical Examples.

1. Having measured a distance of 200 feet from the bottom of a steeple, in a direct horizontal line, I then took the angle of elevation of its top, Abc = 47° 30′. Required the height of the steeple.

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To Ac add the height of the instrument Cc, then Ac+cC AC the height of the steeple.

2. The angle of elevation of a hill was found to be 46°, and 100 yards farther off, on a level with the bottom, it was 31°. Required its height.

ABC 46°- ADC 31° DAB 15o.

=

Sine DAB 15°: sine D 31.:: DB 100: AB.

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3. The angles of elevation of a cloud, or other object, taken by two observers at the same time, on the same side of it, in the same vertical plane, and 880 yards distant from each other, were 64° and 35o. Required its height, and its distance from the two places of observation.

Answer. Height 935.757 yards, greater distance 1631-442, less distance 1041.125.

4. From the top of a tower 120 feet high, which was in a line with two trees on the same horizontal plain as its bottom, I took the angles formed by the perpendicular wall and a line conceived to be drawn from my eye to the bottom of each tree, and found them to be 33° and 64° 30′. Required the distance between the

trees.

Note. An angle taken from the top of an elevated object, usually called the angle of depression, is the angle formed by a straight line conceived to be drawn from the eye to the object and another line parallel to the horizontal plane. Thus, the angle EAB is the angle of depression of the object B, which, by the nature of parallel lines, is equal to the angle ABC (29. 1). The angle BAC is the complement of EAB or ABC, and is the angle used in the calculation.

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R: tan. DAC 64° 30' :: AC 120: DC.

B

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5. I took the angle of elevation of a high inaccessible object, which was 58°, and at the distance of 100 yards farther, in a straight line, I again found the angle of elevation 32°. Required the height of the object, and its distance from the first station, the height of the instrument being five feet. See fig. prob. 4.

The height of the object above the ground is found to be 104-17 yards, as in prob. 2.

R or sine C 90°: cos. B 58° :: AB: BC.

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6. Being desirous to know the height of an obelisk erected on the top of a declivity, I measured from its bottom a distance of 40 feet, and there found the angle of elevation formed by the plain and an imaginary line drawn to the top of the object to be 41°. I then measured 60 feet farther, in the same direction, and found the angle of elevation, formed as before, 23° 45'. Required the height of the obelisk.

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CA + CD 121·488 : CA — CD 41-488 ::

tan. § (A + D) 69° 30′ : tan. 1⁄2 (D— A).

D

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69° 30′ — 42° 24′ 24′′ = 27° 5' 36" = angle CAD.

Sine CAD 27° 5′ 36′′ : sine C 41° :: CD 40: AD.

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line from the hill, he found the elevation CBE 33° 45′ of the top of the castle. Required the height of the castle.

CAE-B ACB= 17° 15'.

=

Sine ACB 17° 15′ : s. B 33° 45′ :: AB 100: AC.

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