9, any triangle and its inscribed circle. The radius K is precisely equal to the radical of (15), whence, if T denote the area of the triangle. T= sK = V b)(s c), a formula attributed to Hero of Alexandria (about 125 B.C.).
Triangle Solution Exemplified.Given a = 32.456, 6 = 41.724, c = 53.987; required the angles. Applying logarithms to (15), log tan TA a = log K log (s a).
s = 64.084, (s a) = 31.628, 6) = 22.36, (sc) = 10.097, log (s a) =1.50007, log (s b) = 1.34947, log (s c) =1.00419, colog s= 8.19325 10.
Summing and taking half, log K = 1.02349; log (s a) = 1.50007; whence log tan =9.52342 10, whence a =18° 27' 23", 54' 46".
In like manner one may find p.50° 32' 32". and y = 92° 32' 44". As a check one finds a + /3 y=-- 180° 0' 2", an excess regarded in practice as very slight and in most work negli gible. To secure more accurate results, which is seldom necessary, it suffices to employ loga rithms of more than five decimal places.
Inverse Trigonometric The symbols arcsin n or n, arccos n or s, arctan n or n, etc., denote respectively an angle whose sine is n, whose cosine is n, etc. They are variously read inverse sine, cosine, etc., of n, or etc., of ts, or, again the arc or angle whose sine, cosine, etc., is ti. They are called inverse trigonometric or circular functions, being related to the direct (so-called) trigonometric or circular functions much as are the integral and the derivative of the Calculus (which see), or the logarithm and the exponential of Algebra (which see). Like analogies abound. It should be noted that etc., do not signify reciprocal of sin, cos, etc. Moreover, unless the contrary is expressed, it is generally understood that n, etc., shall signify the smallest positive one of the infinitely many angles whose sine, etc., is n. Thus will ordinarily mean 30° though, taken in full generality, it would signify 30° ± 2nir, or 150° -I- 2nr, n being any integer. The direct functiOs are one-valued functions of the angle, but the angle is an in finitely many-valued function of a direct function value.
Trigonometric Equations.--These are such as involve one or more direct or inverse trigonometric functions regarded as the un knowns or variables like the x,. y, etc., of ordinary algebra. Such an equation, for ex ample, is sin a + sin 5 a = sin 3 a.
it, apply formula (5); then 2 sin 3 a cos 2 a = sin 3 a; whence either sin 3 a = 0, or 2 cos 2a 1; hence either a = nr: :3 or a = ir :6. For another example let +I= x, to find x. Denote f , and x by a, /9, and y respectively. Then sin a = cos a= A, sin y = x. Also a fi= y, and sin(a,(3)= sine cos cos a sin ft = sin y = x; substituting the values of sin a, etc., it is found that x= if For applications to the solution of the general cubic equation in one un known, the reader is referred to the articles on ALGEBRA Of that 011 EQUATIONS, GENERAL THEORY OF., Some Trigonometric Series. Consider the infinite series al (s) sin a = a - 1.2.3 1.23.4.5 1.2.
> . . . 7 al co (c) coS a = 1 1.2 1.2-3.4 1.2- . . ..8 It may be proved algebraically, and is proved by means of Maclaurin's Expansion (see CALCU LUS), that the series (s) and (c) respectively represent or define sin a and cos a for every finite value of the angle a reckoned in terms of the radian. The precise meaning is that, if Sn denote the sum of the first n terms of (s), then the limit of Sn as n increases endlessly is sin a. Similarly for (c). The algebraic proof is too long for insertion here, and that by the Calculus rests on presuppositions not appropriate in this article. As a compromise it is edifying and interesting to assume the validity of equations (s) and (c) and then after the manner of natural science to test them, regarded as hypotheses by their implications, or conse quences. proof is not thus obtainable, but certainty can be thus more and more nearly approximated. Any consequence of (s) or (c) or both that is known to be untrue would alone suffice to invalidate one or both assumptions absolutely, while any number of consequences known to be true merely tend to support but do not suffice to prove the hypotheses. Some such supporting consequences may be noted. If a = 0 series (s) and (c) become respectively 0 and 1, as should be the case, since sin 0= 0 and cos 0= 1. If a be replaced by a, each term of (s) is reversed in sign, while (c) is unchanged; and this, too, should be so, for, as before seen, the sin is an odd, and the cos an even, function of the angle. Again, it is proved in algebra (see ALGEBRA, also SERIES) that, e being the Napierian base.