Rem. When 0>0 and <π, sin 0 is +, and every factor in the second member of (6) is positive; when > and <2π, sin is, and only the second factor is negative; when >2 and <3 π, both members are positive, since only the second and third factors are negative; and so on. Hence the sign was taken in extracting the square root of (1). -In (2) of Art. 172, 173. Resolve cose into Factors. change into 4 + a, then no becomes no + na, i.e., no+ Hence (2) becomes cos n = 2"-1 sin (+α) sin(+3a) sin (+5α) ф π But sin(+2na − a) = sin(+π − α) = sin (« − p), = sin(+2na-3a) sin (3a), and so on. Hence when n is even we have from (1) cos no 2-1 sin (a + ) sin (a - p) sin (3a + 4) sin (3 α − p) X.. × sin[(n − 1)α + 6] sin[(n − 1)α — $] =2"-1(sin2 asin'p) (sin 3a -: sin2) Therefore, putting no 0, as in Art. 172, we obtain = NOTE. For an alternative proof of the propositions of Arts. 172 and 173, see Lock's Higher Trigonometry, pp. 92-95. 16 cos cos (72°-0) cos (72°+0) cos (144°-0) cos (144°+0) = cos 50. SUMMATION OF TRIGONOMETRIC SERIES. 174. Sum the Series sin a+sin(a+ẞ) + sin(« +2 ẞ) + ... + sin [« + (n − 1) ẞ]. 2 sin (a +2ẞ) sinẞ= cos (a + B) - cos (a + B), etc. etc. 2 sin [a + (n-1)ẞ] sin B Therefore, if S, denote the sum of n terms, we have, by B 2n 2 B = sin[« + 2" - 18] - sin [a + 2" = 3p] 2n 2 Denoting the sum of n terms by S, and adding, we get Rem. The sum of the series in this article may be deduced from that in Art. 174 The sums of these two series are often useful;* and the by putting a + for a. 2 student is advised to commit them to memory. NOTE.- These two results are very important, and the student should carefully notice them. 176. Sum the Series sinTM a+sin"(a+ß)+sinTM (a+2 ß) +...+sin” [«+(n−1)B]. This may be done by the aid of Art. 159 or Art. 160. * See Thompson's Dynamo-Electric Machinery. 3d ed., pp. 345, 346. Thus, if m is even, we have from Art. 159 = (−1)2 [cos m (a+ß) —m cos (m−2) (a+ß) +···] (2) = ... and so on; and the required sum may be obtained from the known sum of the series and [cos ma + cos m (a + B) + cos m (a + 2B) + ...] {cos (m2) a + cos [(m2) (α + B)] +cos [(m-2) (a + 2B)] +}, etc. We may find the sum of the series m cosm α + cosTM (α + B) + cosTM (a + 2 ß) + etc. to n terms in a similar manner by the aid of Art. 158. EXAMPLES. 1. Sum to n terms the series sina+sin(a + B) + sin2 (a + 2B) + ... We have 2 sin2 α = - (cos 2 α-1) by (1), 2 sin2 (a + ẞ) = − [cos 2 (α + B) — 1] by (2), 2 sin2 (a + 2 ẞ) = − [cos 2 (a + 2 ß) − 1], and so on. Hence 2S,=n-[cos 2a+cos 2 (a+ẞ)+cos 2(x+2ẞ) +...] |