hence, the angles of these triangles may be found, and consequently, those of the given triangle.

=

Examples.

1. Given a 40, b = 34, and c 25, to find A, B, and C.

Operation.

Applying logarithms to formula (15), we have

log (s—s') = (a. c.) log (s + s') + log (b+c) + log (b−c)—10;

s' = ↓ (s + s') — § (s — s') = 13.3625.

From formula (11), we find

=

A = 180° 96° 06′ 45′′ = 83° 53′ 15′′.

4, to find A, B

2. Given a = 6, b = 5, and c = 4, and C.

Ans. A 82° 49' 09", B 55° 46' 16", C 41° 24' 35".

=

3. Given a = 71.2 yds., b = 64.8 yds., and c = 37 yds, to find A, B, and C.

Ans. A 84° 01' 53", B = 64° 50' 51", C= 31° 07' 16".

2. At what horizontal distance from a column, 200 feet high, will it subtend an angle of 31° 17' 12"?

3. Required the height of a hill D above a horizontal plane AB, the distance between A and B

being equal to 975 yards,

A

Ans. 329.114 ft.

and the angles of elevation at A and B being respectively

15° 36' and 27° 29'.

Ans. DC 587.61 yds.

are

A

4. The distances AC and BC found by measurement to be respectively, 588 feet and 672 feet, and their included angle 55° 40'. Required the distance AB.

B

5. Being on a horizontal plane, and wanting to ascertain the height of a tower, standing on the top of an inaccessible hill, there were measured, the angle of elevation of the top of the hill 40°, and of the top of the tower 51°; then measuring in a direct line 180 feet

farther from the hill, the angle of elevation of the top of the tower was 33° 45'; required the height of the tower. Ans. 83.998 ft.

6. Wanting to know the horizontal distance between two inaccessible objects E and W,

the following measurements were

7. Wanting to know the horizontal distance between

two inaccessible objects A and

B, and not finding any station

from which both of them could

be seen, two points C and D F

were chosen at a distance from

B

each other equal to 200 yards;

from the former of these points,

A could be seen, and from the
latter, B; and at each of the points C and D a staff was
set up.
From C a distance CF was measured, not in the
direction DC, equal to 200 yards, and from D, a distance
DE equal to 200 yards, and the following angles taken:
ACD = 53° 30′,

AFC = 83° 00', BDE 54° 30',
BDC 156° 25', ACF 54° 31',
Required the distance AB.

BED 88° 30'.

Ans. 345.459 yds.

8. The distances AB, AC, and BC, between the points A, B, and C, are known; viz.: AB = 800 yds., AC = 600 yds., and BC 400 yds. From a fourth point P, the angles APC and BPC are measured; viz.:

Required the distances AP, BP, and CP.

B

This problem is used in locating the position of buoys in maritime surveying, as follows. Three points, A, B, and C, on shore are known in position. The surveyor stationed at a buoy P, measures the angles APC and BPC. The distances AP, BP, and CP, are then found as follows:

Suppose the circumference of a circle to be described through the points A, B, and P. Draw CP, cutting the circumference in D, and draw the lines DB and DA.

The angles CPB and DAB, being inscribed in the same segment, are equal (B. III., P. XVIII., C. 1); for a like reason, the angles CPA and DBA are equal: hence, in the triangle ADB, we know two angles and one side; we may, therefore, find the side DB. In the triangle ACB, we know the three sides, and we may compute the angle B. Subtracting from this the angle DBA, we have the angle DBC. Now, in the triangle DBC, we have two sides and their included angle, and we can find the angle DCB. Finally, in the triangle CPB, we have two angles and one side, from which data we can find CP and BP. In like manner, we can find AP.

ANALYTICAL TRIGONOMETRY.

47. ANALYTICAL TRIGONOMETRY is that branch of Mathematics which treats of the general properties and relations of trigonometrical functions.

DEFINITIONS AND GENERAL PRINCIPLES.

B

Α

48. Let ABCD represent a circle whose radius is 1, and suppose its circumference to be divided into four equal parts, by the diameters AC and BD drawn perpendicular to each other. The horizontal diameter AC is called the initial diameter; the vertical diameter BD is called the secondary diameter; the point A, from which arcs are usually reckoned, is called the origin of arcs, and the point B, 90° distant, is called the secondary origin. Arcs estimated from A, around toward B, that is, in a direction contrary to that of the motion of the hands of a watch, are considered positive; consequently, those reckoned in a contrary direction must be regarded as negative.

The arc AB, is called the first quadrant; the arc BC the second quadrant; the arc CD, the third quadrant ; and the arc DA, the fourth quadrant. The point at which