But as a approaches the limit 0, cos x approaches the limit But each factor approaches the limit 1 as x approaches 0. With O as a centre, and a radius equal to 1, describe the circle AB; draw BD and AE perpendicular to XX', and CF perpendicular to YY'. Then by Art. 32, the functions of AOB are : But since the right triangles OBD, OEA, and OCF are similar, and OA OC= 1, we have, = Whence, since OB = 1, the functions of AOB are: That is, if the radius of the circle is unity, The sine is the perpendicular drawn to XX' from the intersection of the circle with the terminal line. The cosine is the line drawn from the centre to the foot of the sine. The tangent is that portion of the geometrical tangent to the circle at its intersection with OX included between OX and the terminal line, produced if necessary. The cotangent is that portion of the geometrical tangent to the circle at its intersection with OY included between OY and the terminal line, produced if necessary. The secant is that portion of the terminal line, or terminal line produced, included between the centre and the tangent. The cosecant is that portion of the terminal line, or terminal line produced, included between the centre and the cotangent. And with regard to algebraic signs, Sines and tangents measured above XX' are positive, and below, negative; cosines and cotangents measured to the right of YY' are positive, and to the left, negative; secants and cosecants measured on the terminal line itself are positive, and on the terminal line produced, negative. 63. The above are called the line values of the trigonometric functions. They simply represent the values of the functions when the radius is unity; that is, the numerical value of the sine of an angle is the same as the number which expresses the length of the perpendicular drawn to XX' from the intersection of the circle and terminal line. 64. To trace the changes in the six principal trigonometric functions of an angle as the angle increases from 0° to 360°. Let the terminal line start from the position OA, and revolve about O as a pivot in the direction of OY, occupying successively the positions OB.. OR.. OC, OB, OB1, etc. 39 Then since the sine of the angle commences with the value 0, and assumes in succession the values B1D1, BD2, OC, B ̧Ð1⁄2, BD, etc. (Art. 60), it is evident that as the angle increases from 0° to 90°, the sine increases from 0 to 1; from 90° to 180°, it decreases from 1 to 0; from 180° to 270°, it decreases (algebraically) from 0 to -1; and from 270° to 360°, it increases from 1 to 0. Since the cosine commences with the value OA, and assumes in succession the values OD1, OD2, 0, — OD3, — OD1, etc., from 0° to 90°, it decreases from 1 to 0; from 90° to 180°, it decreases from 0 to -1; from 180° to 270°, it increases from 1 to 0; and from 270° to 360°, it increases from 0 to 1. Since the tangent commences with the value 0, and assumes in succession the values AE1, AE2, ∞, - AE3, -AE, etc., from 0° to 90°, it increases from 0 to ; from 90° to 180°, it increases from to 0; from 180° to 270°, it increases from 0 to co; and from 270° to 360°, it increases from - to 0. Since the cotangent commences at, and assumes in succession the values CF11, CF2, 0, −CF3, —CF, etc., from 0° to 90°, it decreases from ∞ to 0; from 90° to 180°, it decreases from 0 to· - ∞; from 180° to 270°, it decreases from ∞ to 0; and from 270° to 360°, it decreases from 0 to -∞. Since the secant commences at OA, and assumes in succession the values OE1, OE2, ∞, OE3, -OE, etc., from 0° to 90°, it increases from 1 to∞; from 90° to 180°, it increases from - to -1; from 180° to 270°, it decreases from −1 to co; and from 270° to 360°, it decreases from ∞ to 1. Since the cosecant commences at ∞, and assumes in succession the values OF1, OF2, OC, OF, OF4, etc., from 0° to 90°, it decreases from ∞ to 1; from 90° to 180°, it increases from 1 to ∞; from 180° to 270°, it increases from ∞ to -1; and from 270° to 360°, it decreases from 1 to -∞. Note. Wherever the symbol occurs in the foregoing discussion, it must be interpreted in accordance with the Note to Art. 41. V. GENERAL FORMULÆ. 65. To find the values of sin (x + y) and cos (x+y) in terms of the sines and cosines of x and y. Let AOB and BOC denote the angles x and y, respectively; then, AOC=x+y. From any point C in OC draw CA and CB perpendicular to OA and OB; and draw BD and BE perpendicular to OA and AC. Since EC and BC are perpendicular to OA and OB, the angles BCE and AOB are equal; that is, BCE: = x. Whence, cos(x + y) = cos x cosy - sin x sin y. = sina sin y. (12) |