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To find the angles.
To find ; sum ang. A & ang. B. As AC+BC 110
180 to ACBC 10
45 Soistan. ang. B+ang. A.67°30' 10.38278
2)135 fum. To tang. ang. B---ang. A 12° 23' 9.34139
180 Given BC 200 Required angles A, C, and AC.
Lang. B 69 °
To find the sum of ang. A&C. To find the angles A and C. 180°
As CB+AB 380 2-57978 69
is to CB-AB 20 1.30103
Soistan.ang. A+C55° 30' 10.16287 2)111 fum 55° 30' half sum of
11,46390 Istotan. ang. A--C4° 23' 8.88412 ang. A and C.
The three sides of any triangle being given, to find the angles.
Fig. 57. plate 4.
AB: AC+BC:: BC. AC:BD-AD
100 : 140 : : 20 : 28
In all cases of this kind, the | To the base
The greater feg.
To find angle A.
To find angle B. As AD 36 1.55630 | As BD 64
1.80618 is to rad 90°
is to rad. 90 So is AC 60 1.77815 So is BC 80
1 To sec. ang. A 53° 8' 10.22185 ) Tofec.ang. B 36° 52' 10.09691
Angle C may be found thus : From 180, subtract the sum of angles A and B, the remainder will give angle C. Or add the complements of the angles A and B together, and the fum is | angle C.
The preceding problem is frequently wrought according to the fol
Add the three fides together, and, from half the sum, subtract the sides severally; then add the complements of the logarithms of the half sum, and of the difference between the halfsum, and the side opposite to the angle sought, to the logarithms of the differences of the two other sides and half sum ; and half their sum will be the tangent of half the angle re. quired. Thus, let angle A be required:
80 I sum 120 half sum 120
Complement 7.92082 the dif. bet. 80 & 120 40 Complement
1.77815 the other differences
53° 8' The angle CAB,
The angles BC may be found by problem 1. of oblique angled trigonometry.
We come now to the application of trigonometry, to the mensuration of heights and distancese
The instruments commonly made use of in measuring heights and distances, are the Geometrical Quadrant, the Theodolite and the Geometrical square.
The Geometrical quadrant is used for investigating vertical angles ; whether they be angles of * altitude, or angles of depression.
The Theodolite serves for measuring angles on a horizontal plane, or on an inclined plane.
A vertical plane, is that which is at right angles with the horizon. A horizontal plane, is that which is parallel to the horizon.
The Geometrical quadrant, is the fourth part of a circle, and is divided into 90°, to which two fights are adapted, and a plumb line fuspended from the centre ; it is commonly made of brass or wood. Fig. 1. plate 4.
N. B. When the object is higher than the measurer's eye, it is said to sube tend an angle of elevation, but when lower, an angle of depresion.