Can anyone prove one or some of these math problems?

Question

With assuming that Z is a complex number(and others are hyperbolic functions):



A) |Sinh y|<= |Sin Z| <= Cosh y

B) |sinh y|<= |cos Z| <= Cosh y

C) |sin Z|^2 + |cos Z|^2 = sinh y ^2 + cosh y ^2


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Thanks

( a<=b, means less or equal)

(hyperbolic functions are those which i inserted "h"after them,as u know,one with out "h"means a normal trigonometry function)

Answer 1

Please check the statement of the problem, then apply this.

We know that
sin (x)=(e^(ix)- e^(-ix))/2i

cos (x)=(e^(ix)+ e^(-ix))/2

also

sin(ix)=i(sinh(x))=(e^(-x)- e^(ix))/2i

cos(ix)=(cosh(x))=(e^(-x)- e^(ix))/2

The only thing you need is if z=x+iy then |z|=sqrt(x^2 + y^2)

Good luck

Answer 2

12

Answer 3

your question is wrong because coshy