SINE AND COSINE. 



SINE-CURE. 



688 



angle. We give, as an instance, the proof of the first formula when 

 both angles are greater than two right angles. Let x o p = a, r o Q = 6, 

 both angles being measured in the direction of revolution indicated by 

 the arrow. The sum is four right angles + x o Q, which has the same 

 sine and cosine as x o Q. From any point Q in o Q draw perpendiculars 

 on o x and o r, and complete the figure as shown. Then sin (a + 1) is 

 positive, and is the fraction which Q M is of Q o, or Q M : Q o ; Q. M and 

 4 o being expressed in numbers. But 



QM 

 qo 



0.3 X R 



go """ QO 



Q S Q N X 11 X O 



Q X Q O >' O Q O 



By similar triangles = + 



NIIW, remembering the magnitudes of a and b, and the rule of signs 

 established, we have 



p T o T <J N N_p 



in a = - ,cosa=- ,sini = - , cos b= - Qo ; 



and substitution immediately gives the first formula. We shall not 

 here dwell on the minor consequences of these formula;, but shall 

 refer to the collection in TRIGONOMETRY. 



The connection of the sine and angle depends in great part upon 

 the following theorem : if x may be made as small as we please, 

 MII .' : x may be made as near to unity as we please. Observe that 

 this theorem supposes the angle x to be measured by the theoretical 

 unit [ANGLE], or the angle 1 to be the angle of which the arc and 

 radius are equal. The proof depends upon the assumption that in the 

 adjoining obvious figure the arc A B is less than its containing contour 



B C 



M A 



A c + c B. If the radius o B be r, we see that x must be arc A B : r, or 

 AB = rx. Also BM = r sin x, BC=AM=r r cos r, by definition. 

 Now the arc A B is greater than B M, and less than B u + M A, or rx liea 

 between r sin x and r sin x + rr cos .T, or x lies between sin x and 

 fin s. + 1 COB x ; 



sin* x 

 or between sin x and sin x + 



whence 



Ues between 



Hence, x diminishing without limit, the difference between 1 and x : 

 sin s diminishes without limit, and therefore that between 1 and sin x 

 : x- which was to be proved. From hence it follows that 1 cos x 

 and {x* approach to a ratio of equality, as may be readily proved from 

 the equation 



x* ( gin x 



*** = 



condition. Hence, making cos 1 f V~ 1 .sin l = e, we have cos 1 

 V 1. sin 1 = e -', and the two equations give 



cos x= 



in which the second and third factors have unity for their limit. 

 Hence then, when x is very small, x and 1 4*' are very near 

 representatives of the sine and cosine; and the goodness of the 

 representation may be increased to any extent by diminishing x. 



The complete theory of the sine and cosine, from and after the two 

 theorems just established, depends upon the introduction of the 

 square root of the negative quantity. If we take ordinary algebra 

 only, in which the impossible quantity is unexplained, we have the 

 most common mode of proceeding. The explanations afterwards given 

 would make this theory the most simple imaginable, to a student who 

 had learned ALGEBRA from the beginning in the manner pointed 

 out. To take the middle course, let us assume the rules of algebra 

 [OPERATION] independently of the meanings of the symbols. Let sin x 

 and cos x be defined as " such functions of x that sin (x + y) gives 

 sin x. cos y + cos x. sin y, and cos (z + y) = cos x. cos y sin x. sin y." 

 Observe that we do not in thus defining say there are such functions ; 

 we only say, if there be such, let them have these names. Then, as in 

 ALGEBRA cited above, we see that if <jw: = cos z+ V 1 sin a:, the 

 relation <p(x + y) = <px x <tj follows ; whence [BINOMIAL THEOREM] <f>x 

 can be nothing but K*, where K is independent of x. Letx=l, 

 whence <p(\) = K, or we have 



cosarf N/-l.sin:r. = (cosl + V !"> 1)' . . . (1) 

 and similarly it a shown that 



cos *-V-l-na: = (cos !/-!. sin 1)* . . . (2). 

 From these we get, by multiplication, 



cos" x + sin' x = (cos' 1 + ein 2 1) : (3) 



if it be possible, let cos 5 1 + sin- 1 = 1, then cos 3 x + sin 5 x is always = 1 , 

 or at least [ROOT] we may always take one. pair of forms satisfying this 



which will be found to satisfy all the conditions used in defining them, 

 namely, 



2 -2 2 " "2 V-l 2 V-l 



e"+ e~*~> e'-e-* e +e~ e* +e~* e c~ 

 2V-1 = 2V-1 

 e* + e-' \ 2 

 2 ) 



To determine what algebraical formula e must be, take the universal 

 formula 



T +( lo 8) 3 9 f -5 + 



whence we easily get from (4) 



loge 



(log e) 3 j~ 

 V-l 2.3 



-r (log e)' T 



(log e) 5 .r 5 

 V-l 2.3.4.5 



(loge) 4 ., 'o , - 



Now e, as far as our definitions have yet extended, is wholly unde- 

 termined, every value of e being applicable. Let us add to our 

 conditions that sin x : x shall approach to unity as x is diminished 

 without limit : but sin x : x approaches to log e : V 1 ; therefore 

 loj? e= V l> or e = ^~*. 



The preceding is purely symbolical ; we merely ask how are pre- 

 vious symbols, used under certain laws, to be put together so as to 

 represent certain new symbols which are to have certain properties. 

 Let us now take the real geometrical meaning of sin x and cos ./;, and 

 the complete system of algebra, in which \/l is explained. In that 

 system, if a line equal to the unit-line be inclined to it at an angle x, 

 it is obviously represented by cos x+ V 1 sm x < and any power of 

 it, whole or fractional, can be obtained by changing x into m.r, so that 



cos mx+ V- 1 sin m.c = (co6 x+ V 1 sin x) M 



is an immediate consequence of definition; and making .r=l, the 

 equation 



cos m+ V 1 B in m = (eoa 1 + V" 1 sin 1)* 



follows at once. To prove that f^~ l and cos 1+ \/l sin 1 are 

 identical, in the most logical manner, requires a previous definition of 

 an exponential quantity, in a sense so general, that exponents of the 

 form a + A v 1 shall be included : without this the new algebra just 

 referred to is not free from the results of INTERPRETATION. 



However we may proceed, the series above given for the sine and 

 cosine of x become 



Sin X X tl o + o > 



2.3 2.3.4.5 ~ 2. 3. 4. 5. C. 7 



x* 

 cos z=l "3" + 



2.3.4 2.3.4.5.8 



and these series are always convergent. Their present form depends 

 entirely on the unit chosen ; if however by x we mean x, x 1 , or x", we 

 must write 



a? .c** a 5 x* 



sin x=ax- .JTJ + 273TT. 5 ~ ' ' ' ' 



cos z=l 



O 5 3? 



2.3.4 



where [ANGLE] o is "01745,32925 ..... "00029,08882, ..... -00000,48481, 



according as x means a number of degrees, of minutes, or of 



seconds. 



The preceding is enough on the fundamental meanings of these 

 terms, and on their connection with algebra. Some applications will 

 be seen in TRIGONOMETBY. 



SINE and COSINE, CURVES OF. By the curve of sines is 

 meant that which has the equation y sin x, and by the curve of 

 cosines, that which has the equation y = cos x ; it being understood 

 that x stands for as many angular units as there are linear units in the 

 abscissa. The undulatory forms of these curves are easily established ; 

 and if the ordinate of a curve consist of several of them, as in 

 y = ft sin x + b cos x + c sin 2x, the several parts of the compound 

 ordinate may be put together in the same manner as that in which 

 the simple undulations are compounded in ACOUSTICS. Except as 

 expressing the most simple form of undulating curves, these equations 

 are of no particular use in geometry. 



SINE-CURE. Sine-cures are ecclesiastical benefices without cure 

 of souls, and are of three sorts : 1. Whure the benefice is a donative 



