CONSTANCY OF FUNCTIONS. THEOR. 1. For a given angle the functions are constant. For, let xop be any angle, P, p', p", ... any points on OP; as and let ap, a'p', A"P", be ordinates of P, P', p", to ox; ... i.e., the same signs, and so for the distances; ... like ratios are equal, both in magnitude and in quality; and so for the other functions. Q. E. D. PERIODICITY OF FUNCTIONS. THEOR. 2. The like functions of congruent angles are identical. For, let be any plane angle; + then ·.· ±2π, #4π, +2nπ stand for one, two,.......... n entire rev ... olutions, forward or backward, ... the terminal line has the same position for the angles and the r's, the x's, the y's may be made identical, each with each, for all the angles thus formed; .. the ratios are identical, sine with sine, cosine with cosine, and so on. Q. E. D. POSITIVE AND NEGATIVE FUNCTIONS. THEOR. 3. For angles in the first quarter all the trigonometric functions are positive. In the second quarter, the sine and cosecant are positive; the cosine, secant, tangent, and cotangent are negative. In the third quarter the tangent and cotangent are positive; the sine, cosecant, cosine, and secant are negative. In the fourth quarter the cosine and secant are positive; the sine, cosecant, tangent, and cotangent are negative. For, if r be taken positive; then ... in the first quarter r, x, y are all positive, [df. in the second quarter r, y are positive, and x is negative, in the third quarter r is positive, and x, y are negative, in the fourth quarter r, a are positive, and y is negative, .. the qualities of the ratios are as given above. Q. E. D. And so they are if r be taken negative; the reader may prove. EXAMPLES. By direct reference to the definitions and by aid of the figures: 1. Tabulate the functions of nπ, i.e., of 0°, 90°, 180°, 2. Tabulate the functions of (n ± })π, i.e., of 30°, 60°, 3. Tabulate the functions of (n±1)π, i.e., of 45°, 135°, .... .... 10. Find sin√3, sin1 cos 50°; cos-1, cos-1 sin 50°. 11. Find tan-1 ±1, tan-1tan; cot-10, cot 'cot &π. with radius 2 draw a circle about o as centre; on the axis or take Q such that oq is 1; through Q draw a parallel to ox, cutting the circle in P, P'; then ... the ordinates of P, P' are 1, and their distances 2, .'. of Xop, xop', and their congruents, the sines are, and there are no other such angles. Make a like construction, with given abscissas and distances, for the anti-cosines; so with given ordinates and abscissas for the anti-tangents and anti-cotangents. There are two primary angles for each function. 14. Show, by geometry, that sin 20 < 2 sin 0, if 0 <0<π. 15. Divide an angle into two parts that shall have their sines in a given ratio, their cosines in a given ratio, .... 16. Construct an angle whose tangent is four times its sine. 17. If π <0<π, show that tan 6> cot 0. 18. Find the primary values for a, ẞ that satisfy the equations sin (3a-2B)=1, sin (4ẞ-a) = . 19. If the radius of a circle be divided in extreme and mean ratio the greater segment is the side of a regular inscribed decagon hence find the functions of 18° and of 36°. § 2. RELATIONS OF FUNCTIONS OF A SINGLE ANGLE. sin2 0 + cos2 0 =1, 1+ tan2 0 = sec2 0, 1+ coť2 0 = csc2 0. i.e., 1+ y2 2.2 2.2 +1= cos2 + sin2 0 =1, 1+ tan2 0 = sec2 0, cot2 0 +1= csc2 0. COR. If be any plane angle, then: and ... cos 01: sec 0 = sin tan cos 0 = tan 0 : √(1 + tan2 0). 1:√(1+tan2 0), |