36<> 



TRIGONOMETRY. 



TRIGONOMETRY. 



370 



metrical functions, and is always used in its old sense, as the line 

 joining the extremities of an arc. 



We shall now make a collection of the principal trigonometrical 

 formula, and properties of the fundamental functions, referring to 

 ANGLE for the modes of measuring angles, to SINE for development of 

 several of the most important points, to SERIES for the expansions of 

 various functions, to MENSURATION and SPHERICAL for the formulas 

 particularly connected with triangles, and to ALGEBRA, ROOT, SUB- 

 SIDIARY ANGLE, &c., for various other usual applications. 



1. No sine nor cosine exceeds unity ; no secant nor cosecant is less 

 than unity ; a tangent or cotangent may have any value ; versed 

 sines and coversed sines are always contained between and 2, both 

 inclusive. 



2. With the sines and cosecants must be remembered the succession 

 + H --- ; with cosines and secants, H --- + ; with tangents and 

 cotangents, + + . Thus when an angle is in the third right 

 angle, or lies between two and three right angles, its cosine is negative, 

 being the third sign of the Succession + +. Versed sines 



and coversed sines are always positive. 



3. With the different functions must be remembered the following 

 aeries of initial values, being those at the beginning of the several right 

 angles : thug 



sin = 0, sin(r'Z) = l, sin 



) = 0, an (3rZ')= -1 



sine I 

 cosecant | 



1 



1 



I cosine I 

 ! secant 



01 



1 



tangent I 

 cotangent | 



oo 



4. To find a function of any number of right angles increased or 

 diminished by a given angle, take the same function, if the number of 

 right angles be even, its cnfunctiun (sine for cosine, cosine for sine, &c.) 

 if the number of right angles be odd : put that sign which belongs to 

 the given function in the right angle to which the whole given angle 

 belongs when the increment or decrement is less than a right angle. 

 Thus we have 



sin ^3 q 9J = COB 9, 



sin ; + 



which is thus obtained : the odd number of right angles (4 repre- 

 senting a right angle) is a direction to put cmine instead of sine, on the 

 i>ppii-<ite side; now jir-0, 9 being less than a right angle, falls in the 

 third right angle, and the sine in that right angle is , so that cos 9 

 mu-t be written. The following results should be remembered : 



siu^g J= cos 8, cos Q - 9J = sin 9, tan (^ - 9j = cot 9; 



9 J = cos 9, cos ^ + 9J = sin 6, tan (5 + 9) = cot 8 ; 



sin (* 9) = sin 9, cos (* 9) - cos 9, tan ( 9) = - ten fl ; 



sin ( + )=- sin 9, cos ( + )=_ cos 9, tan (w + 9) = tan 9 ; 



sin (2ir 9) = - sin 9, cos (2ir 9) = cos 9, tan (2r - 9) = - tan 9 ; 



in ( 9) = sin 9, cos ( 9) = cos 9, tan ( 8) = tan 9. 



5. In the first revolution, 9 and * 9 have the same sines and cose- 



" and 2r-6 the same cosines and secants, 9 and + 9 the same 

 tangents and cotangents. 



6. sin 9 cosec 9 = 1, cos 9 sec 9=1, tan 9 cot 9 = 1, 



sin 9 cos 9 



" = cot 



7. sin- + cos 2 9=1, 1 + ten' 9= sec* 9, 1 + cot 2 9=coaec 2 9. 



1 



tan 9 

 = Vri T tan--0)' cosfl= 



\ ( 1 + tan- 0) ' 



8. If tan 9 = r , then sin = 



9. sin (9 + ft) = sin 9 cos ft + COB 9 sin <J> 

 sin (9 <j>) = sin cos ft cos sin <fi 

 cos (0 + ft) = cos cos ft sin Bin ft 

 cos (9 <(>) = cos 9 cos + sin 9 sin ft. 



ten 9 + ten ft 



10. ten (9 + *)= r_-^T^.,*. tan (9-, 



1 



2 +6r 



tan fl -tan ft 

 1 -|- tan 9 tan <;> ' 



11. sin 9 f Bin cf> = 2 sin 

 sin 9 sin $ = 2 cos 



COS 9 + COB ft = 2 COB 



9 ft 



9 



sin 



9 + <t> 



12. in 29= 2 sin 9 cos 9 



coi 2 9 = con 3 9 gin 2 9=2 cos 3 91 = 1-2 sin' 9, 



ton29 = 2 tan 9-=- (1 tan 2 9). 

 ARTS AJID SCI. DIV. VOL. VIII. 



14. 



/IT *\ 



13. 1 + cos = 2 cos 2 5, 1 + sin = 2 cos 2 ( j ^ j 



/TT 0\ 



1 - cos = 2 sin 2 g, 1 - sin = 2 sin 2 ( j 5- ) 



1 cos 1 sin _ , /T 



1 + cos ~~ ' 2' 1 + sin 9 V4 



sin <(> sin tan ^ (<t> 0) sin ft ^ sin 

 sin ft + sin ~ tan \ (ft + 0) ' cos ft + cos 



15. If be half a right angle, or less, 



cos 9 = J V(l + sin 20) + J V(l sin 2 0) 

 sin = | V(l + sin 2 0) J V(l sin 2 0). 



16. If n be any integer, and if (cos + sin 0)" be developed by the 

 binomial theorem into P + P 1 + P 2 + &c., P being cos" 0, r, being 

 n cos"~' 9 sin 0, &c., then cos n 9 is P f, + p, &c., and sin is 



17. Let c. and s. stand for the cosine and sine of n0 



2 cos 3 0=c, + 1 



4 cos 3 9=c 3 + 3c 1 



8 cos 4 9 =04 + 40, + 3 

 16 cos 5 9=c 5 + 5c J -< 

 32 cos 6 9=c +6c 4 

 64 cos 7 = o, + 7c 5 + 21c~ + 350, 

 1 28 cos 1 * = c, + 8c 8 + 28c, + 56c a + 35 



512 cos'0= c + 10c a + 45c + 120o t + 210c a + 126 



To change these into corresponding formula; for powers of sin 0, 



For c, c., 3 C 4 c 5 o c, C 3 &c. 

 Write s, c, a, C 4 s a B, c a &c. 

 Thus 16 sin 1 9 = 8., 3s., + 10s L 



82 sin 6 9 = - C + 6c, - 15c a + 10. 



18. De Muirre's Theorem and its consequences : 



(cos 9 + sin 9 . -J 1)" = cos + sin . V - 1 

 t'^- 1 = cos 9 + sin 9 . V 1 t-8-S-i = cos 9 sin . V 1 



cos 9 = 



sin 9 = 



2V-1 



If 2 cos = x + - , then 2 V- 1 sin 9 = a; - - 



X X 



and 2 cos n0=z + , and 2 V 1 sm 0=** ~ - 



The versed sine is little used and rarely mentioned in formula; ; and 

 the coversed sine is really only invented for analogy's sake. 



The term line (the Latin word sinus, meaning the bosom) has been 

 the object of much discussion. It was at one time looked on as a bar- 

 barism from the Arabic ; and some endeavoured to substitute semusis 

 intcripta, the half of the chord, for it. Others again thought that it 

 was a corruption of S. Ins., the abbreviation of the above. Dr. 

 Hutton asserts that the Arabic word Jcilt, which is used for the trigo- 

 nometrical sine in that language, also means the bosom in common 

 language ; and we have been told that this is correct ; if so, the Latin 

 inu is only the literal translation of the Arabic. The arc representing 

 a bow (from which it gets its name), half of the string, which repre- 

 sents the sine of half the arc, would come against the breast of the 

 archer. The versed sine (sinus verstu, or turned sine) was called the 

 tagitta, or arrow. The terms tangent and secant are derived in an 

 obvious manner from the old definitions. 



There is little of the history of trigonometry which can be either 

 usefully or intelligibly separated from that of mathematics in general. 

 Up to the middle of the last century it belonged rather to geometry 

 than to algebra ; and even in our own day algebraical trigonometry is 

 not fully established in England, though rapidly making its way. 

 Those to whom trigonometry is only useful as an instrument in the 

 solution of triangles may enjoy the advantage of that specific clearness 

 which geometry gives to the individual proposition in hand, without 

 needing to feel the want of a system which points out the direction of 

 future progress. But those who are to be trained in mathematics for 

 higher views and more difficult applications, must acquire trigonometry 

 in its most algebraical form as a constituent part of the language of 

 algebra, and an element in every step of their future progress. It is 

 worse than useless to attempt, for them, to draw a distinction between 

 algebraical and trigonometrical ; the science will now allow that dis- 

 tinction to remain, and will rather demand new modes of expression 

 than dispense with any of the old ones. 



There are those who feel sensible of incongruity in combining the 

 fundamental notions of space and number together, and would rather, 

 at any expense of trouble, keep them separate, except when th-.iy arc 

 formally united for any particular application. This feeling h, our 

 sympathy ; and if it were possible to present a complete algebra, both 

 in definitions and proce.sses, without recourse to trigonometrical lan- 

 guage, we should willingly agree to the separation. But hitherto it is 



B B 



