CHAPTER III NATURAL SINES, TANGENTS, ETC. Sine W THE sine of any angle acb or the sine of any circular arc ab is the perpendicular distance, as, from one end of the arc a to the radius passing through the other end b of the arc. It is equal to one-half the chord of the arc abn, which is twice the arc ab; or the chord of the arc abn is equal to twice the sine of half the arc, or twice the sine of ab. The sine of the angle tcb, if tcb equals 90°, is equal to the radius of the circle. Cosine t Vangent Cos. Vers. b FIG. 36. The cosine of an arc ab is the distance cs from the center of the circle c to the intersection of the sine as with the radius cb, and is equal to ya or the sine of the arc ta. But the angle tca is equal to the difference between 90° and the angle acb; or the difference between the arcs tab and ab; and is the complement of acb. Hence the cosine of an angle or arc is equal to the sine of its complement, and vice versa. Versed Sine The versed sine of an arc is the distance sb from the foot s of the sine to the arc at b, measured on the radius cb. Natural Sines, Tangents, etc. The versed sine of an arc ab is equal to the rise of twice the arc; or equal to the rise of abn. Tangent The tangent bw of an arc ab is the perpendicular distance from the radius at one extremity of the arc b to the intersection w of the perpendicular bw with the prolongation of a radius drawn through the other extremity of the arc at a. The Secant The secant of an arc is the distance cw from the center of the arc to the intersection of the tangent at w of the prolonged radius ca. If the angle tcb equals 90 degrees and tca be the complement of acb, the sine ya of this complement, its versed sine ty, tangent to and secant co become respectively the cosine, coversed sine, cotangent and cosecant of the angle acb, and vice versa. When the radius ab is equal to unity the corresponding sines, cosines, tangents, etc., are called natural sines, cosines, etc.; and the table containing their lengths for different angles is the table of natural sines, etc. The lengths of the sines, etc., for the arcs of any other circle, whose radius may be greater or less than 1, are found by multiplying the tabular values by such radius. The following table contains only natural sines, tangents and secants; the other lengths may be found for any angle not exceeding 90 degrees as follows: The formula for the solution of the right-angled and the obliqueangled triangle are given; for further information the reader is referred to works on Trigonometry. |