Complex_Convert_1 Consider the complex number \[z={@z1@} \] Evaluate the modulus of \(z\) \((r = |z|)\) and the principal value of the argument of \(z\) (\(\theta = \mathrm{Arg}(z)\)). Enter your results in the correct order, i.e. \([r,\theta]\).

\([r,\theta] = \) [[input:ans1]] [[validation:ans1]]

]]> \[r=\sqrt{x^2+y^2}\] \[r=\sqrt{\left({@x@}\right)^2+\left({@y@}\right)^2}=\sqrt{{@x*x@}+{@y*y@}}\] \[r=\sqrt{@x*x+y*y@}={@rval@}\] \[tan(\mathrm{arg}(z))=\frac{y}{x}=\frac{@y@}{@x@}={@y/x@}\] \[\mathrm{arg}(z)={@argsol@} + k\cdot\pi\] Considering the appropriate quadrant of the complex plane to fix \(k\), we find that \[\mathrm{arg}(z)={@principaltheta@}\].

]]> 1 0 0 argset:[Pi/6,Pi/4,Pi/3]; n1:rand(3)+1; modsol:rand(5)+1; argsol:ev(((-1)^rand(2))*(rand(2)*Pi-argset[n1]),simp); z1:ev(modsol*cos(argsol) + modsol*i*sin(argsol),simp); x:modsol*cos(argsol); y:modsol*sin(argsol); rval:sqrt(x*x+y*y); principaltheta:%pi - mod (%pi - argsol, 2*%pi); [[feedback:Result]]

]]> \(z={@z1@}\) gives \([{@modsol@},{@principaltheta@}]\) 1 0 0 *10 dot 1 i cos-1 lang [ 0 {"author":[{"firstName":"","lastName":"","institution":"Unknown - Legacy question","year":"2025"}],"language":["en"],"isPartOf":"STACK Library","license":"cc-sa-4.0","additional":{"UoE":{"Notes":"This is automatically generated metadata."}},"freeform":{}} ans1 algebraic [modsol,principaltheta] 20 1 0 0 1 1 1 1 1 Result 2.0000000 1 1 0 AlgEquiv ans1 [modsol,principaltheta] 0 = 1.0000000 -1 Result-0-T = 0.0000000 -1 Result-0-F 817828831 1302743320 2023109718 1885837562 987432883 1651673644 927277669 812708432 727047184 1166488693 1719118035 1330127074 1855378840 710572634 2071272006 1565230007 929362269 763010241 572262086 96601302 1065357600 1368599549 2059212278 1989353230 1173992613 1 ans1 [modsol,principaltheta] Result 1.0000000 0.0000000 Result-0-T 2 ans1 [x,y] Result 0.0000000 0.0000000 Result-0-F