IN THE STUDY OF ANALYSIS. 



Z5 



(o.) Cos 6 rr cos' ^& — sin^ ^6; sin ^ zr 2 cos ^d sin ^$. 

 (jp.) Sin (a -\- h) -{' sin (a — b) zz 2 sin a cos 6 ; sin 

 (a -f- i) — sin (a — b) zrz2 cos a sin b. 



(y.) Cos (a — b) -f- cos (a -j- A) zr 2 cos a cos 6 ; cos 

 (a — 5) — cos (a -|- ^) =^ 2 sin a sin i. 



Sin of sum more (or less) sin© of diff. (p) is 2 sin (or cos) X cos. 

 (or sine), &c. 



(70 The following trigonometrical formula} are important 

 in integrations : 



For ^2*«" Co*'' Cos o'6 will go (a) 



2' Hess half way with o bin. 

 Sin. «^°"'<^' just like Co*^- ib) 



but CorS. o6 fbr Cos 6^ repln. (c) 



(a.) 2*°", (pron. twoton) is 2", or 2 to (power) w. Co*° is 

 cosine to (power) o, or Cos° + S o standing here for the next 

 higher number than n, as being the next letter in order alpha- 

 betical : 6 is understood after Co. 



For the expansion of 2" Cosi»+ ^d, Cos (n -f- 1)^ (iCos oS) will 

 go 2d less (pron. toothless), i. e., 



(a.) 2° Cos (n -\- 1} 6 =z Cos (n -\- 1) 6 -]- A Cos (n — 1) tf 

 -j-B Cos (« — 3) ^)-f-"*^^s ff^^^ ^''^'^^ ^ ^*^» ^' ^'i *^® co-efficients 

 1 A B C are the terms of (1 -f- 1)'*+ S the (« + 1) ^ binomial, 

 or o bin series : and this goes just half way y for if (l-j-l)""*" ^ has 

 an even number of terms, the expansion has exactly half of 

 them ; and just half also if (1 -f- 1)°'*"^ ^las an odd number, for 

 the last term of the expansion is then multiplied by half the 

 middle term of the binomial series. 



(6.) Sin**^^'"*"^^ (five syllables) is the same expansion with 

 Co*o, except that Sin to even o or Sin"+' d, (n -f- 1) being 

 even, has its terms Cos (n -f- 1) ^> A Cos (n — 1) 6, &c., writ- 

 ten repin : and Sin to odd o, Sin'^+ * when w + 1 is odd, has 

 the series Sin (n -f 1) ^, A Sin (n — 1) ^, B Sin (n — 3) d, &c., 

 written repin. 



(c.) CorS 0619 (Cos or Sin) (n + 1) 6, as the index « -f- 1 

 is even or odd, the ors in (b) and (c) corresponding as to antece- 

 dents and consequents. ^ 



