The quotient of (168) divided by (169) gives, after dividing the numerator and denominator by cos x cos y cos z, 82. Let x, y and z be any three angles, and from the equations sin y sin (x — z) — sin x sin (y — z) = sin z (sin x cos y If sin z is eliminated, we find cos y sin (x 2) = sin z sin (x − y) cos x sin y) cos x sin (y2) = cos z sin (x − y) These equations may be more elegantly expressed, as follows: sin x sin (y z) + sin y sin (z — x) + sin z sin (x − y) =0 (171) = 0 (172) A number of similar relations may be deduced from these by substituting 90° ±x, &c., for x, &c. 4 cos v cos (v - x) cos (v — y) cos (v — z) = 2 cos x cos y cos z + cos2 x +cos2y+cos2 z — 1 84. The following may be proposed as exercises. (174) sin x+sin y+sin z-sin (x+y+2)= 4 sin(x + y) sin(x+2) sin (y+z) (175) cosx+cos y + cos z+cos (x+y+2) = 4 cos 1⁄2) (x + y) cos 1 (x + z) cos + (y + z) (176) 4 [sin (x+y+2)+2 sin x sin y sin z]2 = 4 [sin (x+y) cos z+ cos (x—y) sin 2] = = = [1 —cos (2 x + 2 y)] (1 + cos 2 z) + [1 + cos (2 x + 2 y)] (1 — cos 2z) +2 (sin 2 x + sin 2 y) sin 2 z = 2 (1+sin 2 x sin 2 y+sin 2 x sin 2 z + sin 2 y sin 2 z-cos 2 x cos 2 y cos 2 z) (179) 85. Let the sum of three angles x, y and z be, or a multiple of x, that is, an even multiple of 2 a condition which is expressed by the equation then, tan (x + y + z) = 0, and the first member of (170) being thus reduced to zero, the numerator of the second number must be zero, or tan x+tan y + tan z = tan x tan y tan z (181) an equation, it must be remembered, that is true only under the condition (180). Since x, y and z may be selected in an infinite variety of ways so as to satisfy (180), it follows from (181) that there is an infinite number of solutions of the problem, "to find three numbers whose sum is equal to their product." Let the sum of three angles x, y and z be or an odd multiple of ; that is, let 2 then, tan (x + y + z) = ∞, and the denominator of (170) must be zero, or tan x tan y + tan x tan z + tan y tan z=1 which, divided by tan x tan y tan z, gives the sums of the first two and of the second two are by (103) 2 cos x cos (y + z) = (− 1)" (cos 2 x + 1) and the sum and difference of these equations are or 4 cos x cos y cos z = − 1)" (cos 2 z + cos 2 y + cos 2 x + 1) 4 cos x sin y sin z = (-1)" (cos 2 z + cos 2 y cos 2 x 1) the upper sign being taken when ʼn in (184) is even, the lower when n is odd. n y is an explicit function of x, and, since x and y are mutually dependent, x is an implicit function of y; but to express x in the form of an explicit function of y, we write* which is read, "x equal to the angle (or arc) whose sine is y," and x is called the inverse function of y, or of sine x. In like manner tan -1 y," &c. y is the angle or arc whose tangent is 88. Many of the formulæ already given may be conveniently expressed with the aid of this notation. Thus, by (16), or if we put y = tan x -1 -1 tany sec ✓ (1+ y2) * This notation was suggested by the use of the negative exponents in algebra. If we have y nx, we also have x = ny, where y is a function of x, and x is the corresponding inverse function of y. The latter equation might be read “x is a quantity which multiplied by n gives y." It may be necessary to caution the be1 ginner against the error of supposing that sin1 y is equivalent to sin y For a general view of the nature of inverse functions, see Peirce's Diff. Calc. Arts. 13, et seq. -1 2 y y1 89. We may also employ the notation sin―1 (cos x) COS 2 tan ~1 (194) or "the arc whose sine is equal to the cosine of x," i. e. "the complement of x"; and sin (cosy) or "the sine of the arc whose cosine is y," &c. We shall have accordingly CHAPTER V. TRIGONOMETRIC TABLES. 90. BEFORE proceeding to the numerical computation of triangles and to other applications of the preceding formulæ, the student should make himself acquainted with the arrangement of, and the mode of consulting, the trigonometric tables. We shall here speak of those points only that are common to all tables, but it will be necessary to consult also the explanations that are always prefixed to a table in order to understand any peculiarity that may attach to it. We suppose also that he is acquainted with the nature and use of the common tables of logarithms of numbers. There are two principal trigonometric tables;* the first, called the Table of Natural Sines, &c., contains simply the numerical values of the sines, tangents, &c. for each given value of the angle; the * The most convenient seven-figure tables yet published in this country are Stanley's, already mentioned, p. 12. Loomis's (New York, 1848) are convenient six-figure tables. Downes's five and four-figure tables, (now publishing in Philadelphia), will be found to possess many advantages of arrangement, type, &c. Computers engaged in extensive and varied calculations require all three of these works, or others equivalent. Trigonometric functions and their logarithms are, for the most part, approximate quantities expressed with more or less accuracy by a greater or less number of decimals; and the selection and use of a particular table in any case is to be determined by the degree of precision sought for in the results. We might indeed employ seven-figure, or even ten-figure tables in all cases, and reject the final figures of our results, when a lower degree of approximation is thought sufficient; but it is clearly a loss of time and labor to employ other figures besides those that are necessary in arriving at the proposed degree of precision. To the above should be added Bowditch's five-figure tables in his Epitome of Navigation, which are doubtless free from all typographical errors, having passed through a number of editions from stereotype plates. Of the foreign tables, we may mention Taylor's, Hutton's, Babbage's, Shortrede's, in England; Callet's, Bagay's, Borda's, in France. Bagay's Tables give the log. functions to every second of the quadrant; Borda's give the functions corresponding to the centesimal division of angles, (Art. 6). For computations requiring more than seven figures, recourse must be had to the ten-figure tables of Vlacq, Thesaurus Logarithmorum Completus, edited by Vega, (Leipzig, 1794). |
