Thursday, 23 July 2015

trigonometry - complex analysis trigonometric inequalities 2


Using the definitions prove that |sinhy||cosz|coshy , |sinhy||sinz|coshy

Conclude that the complex cosine and sine are not bounded in the whole complex plane.




So I used the identity that |cosz|2=cos2(x)+sinh2(y)|sinhy|2=sinh2(y). However, I'm not sure how to show that |cosz|2cosh2y.



Similarly for |sinz| I used the identity |sinz|2=sin2(x)+sinh2(y)|sinhy|2 and because we have the absolute value this implies |sinhy||sinz|. However, again I am unsure how to show that |sinz|coshy.



Also how do these inequalities allow us to conclude that the complex cosine and sine are not bounded in the whole complex plane?

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