8. The distances AB, AC, and BC,

between the points A, B, and C, are

known; viz. : AB

600 yds., and BC

800 yds., AC =

400 yds. From a

fourth point P, the angles APC and

BPC are measured; viz.:

Required the distances AP, BP, and CP.

B

CP = 1042.524 yds.

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. DCB. Finally, in the triangle CPB, we have two angles and one side, from which data we can find CP and BP. ner, we can find AP.

In like man

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

A

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 are DA, the fourth quadrant. The point at which

an arc terminates, is called its extremity, and an arc is said to be in that quadrant in which its extremity is situated. Thus, the arc AM is in the first quadrant, the arc AM' in the second, the are AM" in the third, and the are AM"" in the fourth.

M

B

M

M

49. The complement of an arc has been defined to be the difference between that are and 90° (Art. 23); geometrically considered, the complement of an arc is the arc included between the extremity of the arc and the secondary origin. Thus, MB is the complement of AM; M'B, the complement of AM'; M'B, the complement of AM", and so on. When the arc is greater than a quadrant, the complement is negative, according to the conventional principle agreed upon (Art. 48).

The supplement of an arc has been defined to be the difference between that are and 180° (Art. 24); geometrically considered, it is the arc included between the extremity of the arc and the left-hand extremity of the initial diameter. Thus, MC is the supplement of AM, and MC the supplement of AM". The supplement is negative, when the arc is greater than two quadrants.

51. The cosine of an arc is the distance from the secondary diameter to the extremity of the arc: thus, NM is the cosine of AM, and N'M' is the cosine of AM'.

The cosine may be measured on the initial diameter: thus, OP is equal to the cosine of AM, and OP' to the cosine of AM'; that is, the cosine of an arc is equal to the distance, measured on the initial diameter, from the centre of the arc to the foot of the sine.

52. The versed-sine of an arc is the distance from the sine to the origin of arcs: thus, PA is the versed-sine of AM, and P'A is the versed-sine of AM'.

53. The co-versed-sine of an arc is the distance from the cosine to the secondary origin: thus, NB is the coversed-sine of AM, and N'B is the co-versed-sine of AM".

54. The tangent of an arc is that part of a perpendicular to the initial diameter, at the origin of arcs, included between the origin and the prolongation of the diameter drawn to the extremity of the arc: thus, AT is the tangent of AM, or of AM", and AT"" is the tangent of AM', or of AM"".

55. The cotangent of an arc is that part of a perpendicular to the secondary diameter, at the secondary origin, included between the secondary origin and the prolongation of the diameter drawn to the extremity of the arc: thus, BT' is the cotangent of AM, or of AM", and BT" is the cotangent of AM', or of AM"".

56. The secant of an arc is the distance from the centre of the arc to the extremity of the tangent: thus, OT is the secant of AM, or of AM", and OT" is the secant of AM', or of AM"".

57. The cosecant of an arc is the distance from the centre of the arc to the extremity of the cotangent: thus, OT' is the cosecant of AM, or of AM", and OT" is the cosecant of AM', or of AM".

The prefix co, as used here, is equivalent to complement; thus, the cosine of an arc is the "complement sine," that is, the sine of the complement, of that arc, and so on, as explained in Art. 27.

The eight trigonometrical functions above defined are also called circular functions.

RULES FOR DETERMINING THE ALGEBRAIC SIGNS OF CIRCULAR FUNCTIONS.

58. All distances estimated upward are regarded as positive; consequently, all distances estimated downward must be considered negative.

Thus, AT, PM, NB, P'M', are positive, and AT", P"M"", P"M", &c., are negative.

All distances estimated toward the right are regarded as positive; consequently, all distances estimated toward the left must be considered negative.

T"

B

M

N

N'

P

Thus, NM, BT', PA, &c., are positive, and N'M', BT", &c., are negative.

P'

N"

These two rules are sufficient for determining the alge braic signs of all the circular functions, except the secant and cosecant. For the secant and cosecant, the following is the rule:

All distances estimated from the centre in a direction toward the extremity of the arc are regarded as positive;