ANALYTICAL TRIGONOMETRY.

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

DEFINITIONS AND GENERAL PRINCIPLES.

C

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 estinated from A, around towards 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

arc terminates, is called its extremity, and an arc is said

an

B

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 arc AM" in the third, and the arc AM"" in the fourth.

M

M

M

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

between the extremity Thus, MB is the complement of AM';

The supplement of an arc has been defined to be the difference between that arc 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 M"C 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 NM' 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'.

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 diam eter through 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 through 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 se cant of AM', or of AM"".

57. The cosecant of an urc 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 term co, in combination, is equivalent to complement of; thus, the cosine of an arc is the same as the sine of the complement of that arc, the cotangent is the same as the tangent of the complement, and so

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

RULES FOR DETERMINING THE ALGEBRAIC SIGNS OF CIRCULAR

FUNCTIONS.

58. All distances estimated upwards are regarded as pos itive; consequently, all distances estimated downwards must be considered negative.

Thus, AT, PM, NB, P'M',

Thus, NM, BT", PA, &c.,

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

All distances estimated from the centre in a direction to towards the extremity of the arc are regarded as positive ; consequently, all distances estimated in a direction from the second extremity of the arc must be considered negative.

Thus, OT, regarded as the secant of AM, is estimated in a direction towards M, and is positive; but OT, re

garded as the secant of AM", is estimated in a direction from M", and is negative.

These conventional rules, enable us at once to give the proper sign to any function of an arc in any quadrant.

59. In accordance with the above rules, and the definitions of the circular functions, we have the following principles:

The sine is positive in the first and second quadrants, and negative in the third and fourth.

The cosine is positive in the first and fourth quadrants, and negative in the second and third.

The versed-sine and the co-versed-sine are always positive.

The tangent and cotangent are positive in the first and third quadrants, and negative in the second and fourth.

The secant is positive in the first and fourth quadrants, and negative in the second and third.

The cosecant is positive in the first and second quadrants, and negative in the third and fourth.

LIMITING VALUES OF THE CIRCULAR FUNCTIONS.

60. The limiting values of the circular functions are those values which they have at the beginning and end of the different quadrants. Their numerical values are discovered by following them as the arc increases from 0° around to 360°, and so on around through 450°, 540°, &c. The signs of these values are determined by the principle, that the sign of a varying magnitude up to the limit, is the sign at the limit. For illustration, let us examine the limiting values of the sine and tangent.