TRIGONOMETRY. CHAPTER I. FIRST PRINCIPLES OF PLANE TRIGONOMETRY. 81. TRIGONOMETRY is that science which investigates the relations of the sides and angles of triangles. When confined to plane triangles, it is called Plane Trigonometry. Definition of an Angle. 82. When two straight lines, AB, AC, meet each other, the opening between these lines is called an angle. This angle or opening, which is read BAC or CAB, or simply the angle at A, does not depend upon the lengths of these lines, but only upon the difference of their directions. C B. Ci Ce B C8 angle CAC; and the same for other arcs and their corresponding angles. Hence, the arcs thus described may be taken as measures of their corresponding angles. §4. When the line AC, has reached the position AC, so that the arc CC3 is a quadrant, or one-fourth of the entire cir cumference, the angle CAC, measured by this quadrant, is called a right angle. The right angle is usually taken as the unit angle; and all angles less than a right angle are called acute angles, and are each measured by an arc less than onefourth of an entire circumference. When the line AC, has reached the position AC, the point Co has passed over an arc equal to one-half of the entire circumference, or over an arc which is the measure of two right angles. Hence, when two lines meet from opposite directions, they are said to form an angle equal to two right angles. An angle such as CAC, which is greater than a right angle and less than two right angles, is called an obtuse angle. 85. All the foregoing relations hold good, whatever be the length of the radius ACo. For simplicity, we shall hereafter use such arcs, for the measure of their corresponding angles, as are described with a unit radius. The Greek letter is usually employed to denote the semicircumference of a circle whose radius is a unit, or, which is the same thing, it represents the ratio of any circumference to its T diameter. Hence is the length of an arc called a quadrant, 2 which is the measure of a right angle. The numerical value of is as follows: T=3.1415926535897932384, &c. semi-circumference, when the radius=1. (See Geom. B. IV., Lemma.) §6. If the whole circumference be divided into 360 equal parts, one of those parts is called a degree. Degrees are also divided into 60 equal parts, called minutes; minutes are similarly divided into 60 equal parts, called seconds, and so on for other subdivisions. Thus 43° 14′ 12′′ denotes an angle whose measuring arc consists of 43 degrees, 14 minutes, and 12 seconds. In the same way 90° denotes an angle whose measuring arc consists of of 360 degrees, and whose length is therefore 2 = 1.5707963267948966192, &c. From the above value of, which is the arc of 180°, it is easy to find the length of an arc corresponding to any angle expressed in degrees, minutes, and seconds. Thus, we find, |