122. The tangent of a very small angle is equal to the angle itself very nearly; the angle being expressed by Now when is very small, tan is very small, and therefore (tan0)2, and all higher powers of tan 0, may be neglected when compared with 1; have ..tan 0, nearly, when is very small. COR. Hence, if a and ẞ be two very small angles, we (Cos a + √ - 1 sin a). (cos ẞ + √ 1 sin ß) cos ẞ-sin a. sin ẞ) +√ −1 . (cos a. sin ẞ+cos ß. sin a) = cos (a + B) + √−1 . sin (a + ß)...... Suppose this law to hold for n factors, so that .(1). (cos a +1. sin a). (cos ẞ+√1. sin ẞ)... (cos +√1. sin x) -1. = cos (a + B + ... + k) + √ −1 . sin (a + ß + ... + k) ; then by introducing another factor, we have (cosa +1. sina)... (cosk+√1. sink). (cos +√1. sin X) = {cos(a+B+...+k)+√√√ −1. sin (a+ß+...+k)}. (cosλ+√/−1. sinλ) = cos (a + ß + ... + λ) + √ −1 . sin (a + ß + ... + λ), by (1) ; and therefore the law holds for n + 1 factors. = But by actual multiplication the law has been shewn to hold for two factors, it must therefore hold for three factors, and thus by successive inductions we conclude that it holds generally. Let S1 = the sum of the quantities tan a, tan ß, ... S2 = the sum of the products of every two of them, S3 = the sum of the products of every three of them; and so on. Then cos (a + B+ +λ) + √1. sin (a + ß + ... ... (cos a +1. sin a). (cos ß + √−1 . sin ß)........ (cos λ + √-1. sin λ) = cosa. (1+√-1.tana).cosẞ. (1+/-1.tan ẞ)...cosλ. (1+/-1.tanλ) = cosa.cos ẞ...cosλ. (1 + √1. S1 - S1⁄2 − √ −1 . S1⁄2 + S1 + ...). . S4 Wood's Algebra, Art. 271. Hence, equating the possible and impossible parts, we have Cos (a + B +...+λ) = cosa. cosẞ...cosλ. (1 - S2+ S1-S6+...) Sin (a+B+...+λ) cosa. cos B...cosλ. (S1- S3+ S5 - ...) ; = ... Tan (a+ẞ+...+λ) = sin (a+B+...+λ) _ S1 - S3+ S5 - ... = If there ben of the angles a, ß, ... λ, it may easily be shewn, as in Art. 112. Cor. that 124. If sin p If sin p = sin P. sin (z + p), required to expand p in terms of sin P and of the sines of z and its multiples. the equation becomes by substituting such values of Sin Ρ Sin (x+p), and and multiplying both sides of the equation by 2√1.e? √ —1‚ we have - 2 ..2√(1 sin P. e*√ 1) = 1 − sin P. € ̃*√1 ; .. 2p √√ − 1 = ], (1 − sin P. e- √ ‒ 1) − ], (1 − sin P. e*√ −1) 22 32 + sin P. ~ √ 1 + 1 (sin P)°. €22 √ −1 + } (sin P)3. €3* √ −1 + ... = sin P. sin x + (sin P)2. sin 2≈ + (sin P)3. sin 3x + ... P 125. In the same manner if tan l = n. tan l, and the tangents be expressed in terms of l√√ - 1 and l' √ - 1 by the formula of Art. 118, it may be proved that, l' = l - m. sin 21 + m2. sin 41 - m3. sin 67+... 126. Required the number of seconds contained in the angle 0, which is expressed by the measure arc radius Ex. 1. Thus the number of seconds contained in the angle p, which, Art. 124, is expressed by the measure arc is radius [The angle l, which is the observed latitude of a place, is read off in degrees, minutes, and seconds from the instrument by which the observation is made; therefore to find the degrees, minutes, and seconds in l', we have only to determine the number of seconds in the latter part of the series, viz. of 0 and its multiples; e being less than 1. By hypothesis, e is less than 1; and since (1b)2, or 1 − 2b + b2, being a square, is necessarily positive, 1+b2 is greater than 2b. Let therefore |