参考答案

正确答案：(1)sin z=sin(x+iy)=sinxcos(iy)+cos xsin(iy)=sin xcosh y+icos xsinh y．(2)cos z=cos(x+iy)=cosxcos(iy)一sin xsin(iy)=cos xcosh y+isin xsinh y．(3)｜sin z｜²=｜sin xcosh y+icos xsinh y｜²=sin² xcosh² y+cos² xsinh²y=sin²x(cosh²y—sinh²y)+(cos²z+sin² x)sinh²y=sin²z+sinh²y．(4)｜cos z｜²=｜cos xcosh y—isin xsinh y｜²=cos²xcosh²y一sin²xsinh²y=cos²x(sinh²y+1)+sin²xsinh²y=cos²z+cos²xsinh²y+sin²xsinh²y=cos²x+(cos²x+sin²x)sinh² y=cos²xsinh²y．
(1)sinz=sin(x+iy)=sinxcos(iy)+cosxsin(iy)=sinxcoshy+icosxsinhy．(2)cosz=cos(x+iy)=cosxcos(iy)一sinxsin(iy)=cosxcoshy+isinxsinhy．(3)｜sinz｜2=｜sinxcoshy+icosxsinhy｜2=sin2xcosh2y+cos2xsinh2y=sin2x(cosh2y—sinh2y)+(cos2z+sin2x)sinh2y=sin2z+sinh2y．(4)｜cosz｜2=｜cosxcoshy—isinxsinhy｜2=cos2xcosh2y一sin2xsinh2y=cos2x(sinh2y+1)+sin2xsinh2y=cos2z+cos2xsinh2y+sin2xsinh2y=cos2x+(cos2x+sin2x)sinh2y=cos2xsinh2y．