Ernest William Hobson

“ The unit of circular measure is too large to be convenient for practical purposes, but its use introduces a simplification into the series in analytical trigonometry, owing to the fact that the sine of an angle and the angle itself in this measure, when the magnitude of the angle is indefinitely diminished, are ultimately in a ratio of equality. ”

“ The trigonometrical functions are employed in many branches of mathematical and physical science not directly concerned with the measurement of angles, and hence arises the importance of analytical trigonometry. The solution of triangles of which the sides are geodesic lines on a spheroidal surface requires the introduction of other functions than those required for the solution of triangles on a plane or spherical surface, and therefore gives rise to a new branch of science, which is from analogy frequently called spheroidal trigonometry. ”

“ Euler's great improvement consisted in his regarding the sine, cosine, &c., as functions of the angle only, thereby giving to equations connecting these functions a purely analytical interpretation, instead of a geometrical one as heretofore. The exponential values of the sine and cosine, De Moivre's theorem, and a great number of other analytical properties of the trigonometrical functions are due to Euler, most of whose writings are to be found in the Memoirs of the St Petersburg Academy. ”

“ The decimal system measurement of angles has never come into ordinary use. In analytical trigonometry the circular measure of an angle is employed. In this system the unit angle is the angle subtended at the centre of a circle by an arc equal in length to the radius. ”

“ Each time that the straight line makes a complete revolution round we consider it to have described four right angles, taken with the positive or negative sign according to the direction in which it has revolved ”

“ Let H denote the radius of the circumscribed circle, r of the in- Radii of scribed, and r lt r 2 , r 3 of the escribed circles of a triangle ABC ; the circum- values of these radii are given by the following formulae. _ _ a be _ a ~ = n r=- = (s-a) tan A = s Of sin B sinj C, = - = s tan A = iB sinj A cosj B cos J C. s a scribed, in scribed, and escribed circles of a tri angle. Spherical Trigonometry. We shall throughout assume such elementary propositions in spherical geometry as are required for the purpose of the investiga tion of formulae given below. ”

“ The sin A sin B sin C equation a = b cos C+c cos 5 expresses the fact that the side a is equal to the sum of the projections of the sides b and c on itself ; thus we obtain the equations b=c cosA + acosC -. c=acos5+6cos^ J If we multiply the first of these equations by -a, the second by b, and the third by c, and add the resulting equations, we obtain the formula Z> 2 + c 2 -a 2 =2&ccos^ or cos A= - - , which gives the cosine of an angle in terms of the sides. ”

“ TRIGONOMETRY 571 connected with the circle. We may easily show from the definitions that cos 2 (a; + iy) + sin* (x + iy) = 1, cosh 2 y - sinh - y 1 ; cos (ar + iy) = cos x cosh y - 1 sin x sinh y, sin (x + iy) = sin x cosh y + 1 cos x sinh y, cosh (a + |3) = cosh a cosh /3 + sinh a sinh /3, sinh (a + /3) = sinh a cosh - cosh a sinh /3. These formulae are the basis of a complete hyperbolic trigonometry. The connexion of these functions with the hyperbola was first pointed out by Lambert. Expan- If we equate the coefficients of 71 on both sides of equation (13) , sionofan we set . . ”

“ We have thus De Moivre s theorem that cos + 1 sin k6 is always one value of (cos + 1 sin 0) *, where k is any real quantity. The principal object of De Moivre s theorem is to enable us to The n c j 11 ,, , , roots of a nnd all the values ot an expression of the form (a + ib) n , where m com piex and n are positive integers prime to each other. If a=rcos0, auan titv m m * " 6 = rsin0, we require the values of r" (cos + t sin 0) ". One value is immediately furnished by the theorem ; but we observe that, sincn the expression cos + tsin is unaltered by adding any multiple of o . ”

“ It has been shown by Remy (in Crete s Journ., vol. iii.) that for any quadrilateral, if z be the spherical distance between the middle points of the diagonals, cos a + cos b + cos c + cos d = 4 cos Jar cos y cos z. ”

“ Math. Journ., vol. iv., in which the series for cos 7i0, sin0 in ascending powers of cos and sin are extended to the case of a fractional value of n. These series have been considered by Euler in a memoir in the Nova acia, vol. ix., by Lagrange in his Calcul d?s fonctioiis (1806) , and by Poinsot in Rccherchcs sur Vatuilyse des sec tions angulaircs (1825) . The general definition of Napierian logarithms is that, if 6*+^ Theory = a + ib, then x + iy= log (a + ib) . Now we know that e x+t y= e*cos y y= ic c siny; hence e? ”