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|2≤cosh2y.
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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