sin-Z-100-find-z-

Question Number 6908 by Tawakalitu. last updated on 02/Aug/16

s i n (Z) = 100

f i n d z

Commented by Rasheed Soomro last updated on 02/Aug/16

A l w a y s

- 1 ⩽ s i n (z) ⩽ 1 \forall z \in R, s o s i n (z) = 100 i s n o t p o s s i b l e .

I n o t h e r w o r d s n o r e a l s o l u t i o n .

Answered by sandy_suhendra last updated on 02/Aug/16

Z i s a c o m p l e x n u m b e r = x + i y

s i n (Z) = s i n (x + i y)

100 = s i n x . c o s h y + i c o s x . s i n h y

100 + 0 . i = s i n x . c o s h y + i c o s x . s i n h y

S o \Rightarrow (1) c o s x . s i n h y = 0

w e g e t, c o s x = 0 \Rightarrow x = \frac{π}{2} + k . 2 π a n d s i n x = \pm 1

o r s i n h y = 0 \Rightarrow y = 0, t h i s i s n o t s o l u t i o n b e c a u s e (x + i y) b e c o m e s r e a l

a n d \Rightarrow (2) s i n x . c o s h y = 100, s u b s t i t u t e s i n x = \pm 1

c o s h y = \pm 100 \Rightarrow y = \pm a r c c o s h 100

S u b s t i t u t e x a n d y t o Z = x + i y

= (\frac{π}{2} + k . 2 π) \pm i a r c c o s h 100

Commented by Tawakalitu. last updated on 02/Aug/16

W o w \dots T h a n k s f o r y o u r h e l p

Commented by Yozzii last updated on 03/Aug/16

I f y \in R, c o s h y ⩾ 1 > 0 \Rightarrow ∄ y \in R f o r c o s h y = - 100 .

I n f a c t, s i n (\frac{π}{2} + 2 k π) = s i n \frac{π}{2} = 1 f o r a l l k \in Z

d u e t o p e r i o d i c i t y o f s i n u s o i d a l f u n c t i o n .

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