# Angles and Triangles (Zambak) by Cihan Mert, Mehmet Özüm

By Cihan Mert, Mehmet Özüm

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Extra resources for Angles and Triangles (Zambak)

Sample text

69) 49 THE REMARKABLE SINE FUNCTIONS 50 Analogously, formula for cosh

54) (55) It i s e a s y t o s e e t h a t f o r m u l a s ( 5 1 ) follow f r o m t h e s e l a s t f o r m u l a s . F o r e x a m p l e , f r o m f o r m u l a s ( 5 2 ) and ( 5 3 ) a p p l i e d s u c c e s s i v e l y , we o b t a i n sin (—i) = sin[/(/i)] = /sinh(/i) = / · / s i n i = —sini. 17. T h e r e s u l t s of s e c t i o n s 1 4 and 1 5 c a n b e o b t a i n e d in a v e r y s i m p l e m a n n e r if we a s s u m e t h a t t h e f a m i l i a r r u l e of c h a n g e of v a r i a b l e u n d e r t h e i n t e g r a l s i g n i s a l s o a p p l i c a b l e t o i n t e g r a l s of functions of a c o m p l e x v a r i a b l e .

C l a ( l + s F a ) ' ^ ' T o e s t a b l i s h t h e addition t h e o r e m f o r t h e l e m n i s c a t e c o s i n e a s s i m p l y a s p o s s i b l e , l e t u s f i r s t r e p l a c e / in ( 8 2 ) with ö)/2— W e obtain c\{^-t) = s\t. (82') F r o m ( 8 2 ) , it follows t h a t cl (a + β) =^ si [(I - a ) - β] = si [ ( | - β) - a]. L e t u s r e p l a c e α by ω/2 —α and β by - β in t h e f i r s t e x p r e s ­ s i o n s ( 8 7 ) and l e t u s r e p l a c e α by ω / 2 —β and β b y — α in t h e s e c o n d e x p r e s s i o n .