is the form of the general quadratic factor of x2+1. When n is even and =2m there are m such factors. When n is odd and 2m+1 there are m such factors; the remaining factor is x+1, as is clear from the figure. EXAMPLES. IX. (1) Find the roots of the equation x2+1=0, and write down the quadratic factors of 4+1. (2) Write down the quadratic factors of x6 +1. (3) Write down the general quadratic factor of x2o +1=0. (4) Find all the values of 1. (5) Find the factors x13+1. (6) Find a general expression for all the values of -1. where scr stands for the sum of the products of the sines taken r together each multiplied by the product of the remaining n-r cosines. (2) With the notation of Ex. 1, prove that cos (a1+a2+az n terms)=cn- Cn−2$2 + Cn−484 — etc. ... (3) Write down the expansion of sin (a+B+y+d+e) and of cos(a+B+y+d+e). (4) Prove that in the series of expressions formed by giving to r in ( COS 0+2rπ in succession, the product of any two equidistant from the beginning and the end is constant. 15 (5) One value of (√3+ √-1) is 27 (√-1+1). (6) From the identity + + (x-b)(x-c)(x-c) (x − a) (x − a) (x —b) ̧ sumptions for a, b and c that sin (0-8) sin (0-y) deduce by assuming x-cos 20+i sin 20, and corresponding as two similar expressions=0. (7) Prove that the n nth roots of unity form a series in G. P. CHAPTER III. RESULTS OF DE MOIVRE'S THEOREM. 18. We proceed to deduce many important results from De Moivre's Theorem. We shall generally in this chapter write i for √-1. 19. By Art. 12, when n is an integer we have Expand the right-hand side of this identity by the binomial theorem, remembering that -1 and that i=+1. Equate the real part of the result to cos no. This gives us cos no cos"0 n (n − 1) = cos"-20. sin'0 12 + n (n − 1) (n − 2) (n − 3) |4 cos"-40. sin10 - etc. Equate the imaginary part to i sin no. This gives us 20. In the above n is a positive integer, and the last terms in the series for cos no and for sin ne will be different according as n is even or odd. (1) sin 40-4 cos30. sin 0-4 cos 0. sin3 0. (2) cos 40=cos1 0-6 cos20. sin20+ sin1 0. (3) The last term in the expansion of cos 100 is - sin1o 0. (4) The last term in the expansion of sin 120 is - 12 cos. sin11 0. (5) When n is even the last term in the expansion of cos ne is n (-1) sin" 0. (6) When n is odd the last term in the expansion of cos ne is EXPONENTIAL VALUES OF SINE AND COSINE. 21. By De Moivre's Theorem, when n is any commensurable number, and x any angle, (cos nx + i sin nx) is a value of (cos x + i sin x)”. For x put the unit of angular measurement; then (cos n + i sin n) is a value of (cos 1 + i sin 1)”. Let k stand for (cos 1 + i sin 1), then (cos n + i sin n) is a value of k", where k is independent of n. Whatever other values (cos 1+ i sin 1)" may have, in what follows we shall only use the value (cos n + i sin n). 22. This important result is a symbolical statement of the fact that expressions of the form cos n + i sin n are combined by the laws of indices. 23. Let the unit of angle be a radian. [E. 59.] [Art. 21.] 1 = 2 {log + (log) + etc.}. [Art. 3.] 13 where R is finite for all values of (since sin 0 is always less than 0, and.. log.k is finite). Let be infinitely diminished. Then, since is the sin 0 circular measure of the angle, the limit of is 1. [E. 290.] Also the limit of the right-hand side is log.k. Therefore, when is the circular measure of the angle, cos + √ 1 sin 0 = e√—1o ̧ i0 =e -io 24. Since cos + i sin =e and cos i sin = e ; are exponential values of the cosine and sine respectively, when the angle is expressed in circular measure. These results may be applied to prove any general formula in Elementary Trigonometry. prove the Use the exponential values of the sine and cosine to following: (1) cos2a+sin2a=1. (2) cos 2a = cos2 a - sin2 a. a cos 0=cos ( − 8). - sin2 ß=cos2 ß – sin2 a. (7) sin 30-3 sin 0 - 4 sin3 0. (5) cos (a+B). cos (a — ß)=cos2 a |