A canon of secants and cosecants. Formulæ of between 1' and 45o, we may form the tangents between 45o and Thus if we replace successively by 1', 2′, 3'..., we get tan 45o. 2′ = 2 tan 4' + tan 44°. 58′, tan 45o. 3′ = 2 tan 6′ + tan 44o. 57′, 804. A canon of secants and cosecants may be formed immediately from a canon of cosines and sines, the secant being the reciprocal of the cosine, and the cosecant the reciprocal of the sine: or much more rapidly, from a canon of tangents and cotangents by means of the formulæ 805. Formulæ of verification are equations between the sines and cosines, tangents and cotangents of different angles, which if satisfied by their values, as given in the canon, or not, would verify their correctness, or the contrary. Such formulæ, however, are better suited to verify the correctness of canons of sines and cosines, tangents and cotangents, already formed and calculated, than to aid us in providing against the intrusion and transmission of errors in the progress of their formation. These equations are easily verified by developing the sines and cosines of the sums or differences of the angles which they involve, and substituting the numerical values of the sines and cosines of 30° and 18°, or their multiples. Demoivre's formula. CHAPTER XXX. ON DEMOIVRE'S FORMULA, AND THE EXPRESSION, BY MEANS OF IT, OF THE ROOTS OF 1 AND OF a±b√−1. 806. IF we multiply together two such expressions as cos + a sin and cos 0 + a sin 0, we shall find (cos + a sin 4) (cos 0 + a sin 0) = cos cos 0 + a2 cos o cos 0 + a (cos & sin 0 + sin cos ). If we replace a by-1, and therefore a2 by - 1, this equation becomes Ꮎ (cos 30+√1 sin 30) (cos +√1 sin 0) = (cos +1 sin 0)* The law of formation of this formula being thus indicated, we may assume 0 (cos +1 sin 0)-1= cos (n - 1) 0+1 sin (n-1) 0; and making = (n − 1)0, and therefore +0=n0, we get {cos (n − 1)0+ √- 1 sin (n − 1) 0} (cos + √√ — 1 sin @) = cos no + √1 sin ne = (cos +/- 1 sin 0)-1 (cos + √1 sin@) It thus appears, that if the formula cos (n − 1) 0 + √ 1 sin (n − 1) 0 = (cos 0 + √√ — 1 sin @)^-1 be true, then the formula is necessarily true (Art. 447): and inasmuch as it has been shewn to be true when n is 2, 3, 4, it is necessarily true when n is 5, 6,... and so on, for any whole number whatsoever. In a similar manner it may be shewn that cos no - √ 1 sin n0 = (cos 0 - √1 sin 0)". Again, if, in the formulæ just established, we replace by m (for may be any angle, great or small, positive or negative*), we get (cos +1 sin 0) (cos 0-√1 sin 0) = cos2 0 + sin2 0 = 1, Exponential expression for the sine and cosine of 0. cos n0 ± √√− 1 sin 0 = (cos 0 ± √ — 1 sin 0)′′* is true for all values of the index. It is known, from the name of its discoverer, as Demoivre's formula, and constitutes one of the most important propositions in the whole range of analysis. 807. If, in virtue of the preceding proposition, we should suppose a to be equal to cos 0+√-1 sin 0, and therefore are: = cos n0+ √√1 sin n0, we might treat are and cos no + √− 1 sin no as possessing common properties, and as immediately convertible with each other: we should thus find The demonstration in the text is dependent upon the properties of indices, by the aid of which we are enabled to shew that cos no±√√1 sinne and (cos±√1 sin 0)a are in every respect convertible into each other and as the general properties of indices are referrible for their authority to the "principle of the permanence of equivalent forms," (Art. 631), so likewise must the formula under consideration be ultimately referrible to the same principle for its establishment: but the properties of indices being once admitted as algebraical truths, we refer to them as furnishing the immediate authority for other truths deducible by means of them, and not to the fundamental principles upon which they rest in common. |