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Find modulus and argument of two complex numbers.
Then use De Moivre's Theorem to find powers of the complex numbers.
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "Use de Moivre's theorem to write the following complex number in the form $a+bi$.
\nRemember that the argument of a complex numbers lies in the range $-\\pi \\lt \\theta \\le \\pi$.
\nImportant
\nWhen calculating the final answer in part (iii), you must use whole number values (integers), otherwise the final answer will not be marked as correct.
", "advice": "Given a complex number $z=r(\\cos(\\theta)+i\\sin(\\theta))$ de Moivre's theorem states that $z^n=r^n(\\cos(n\\theta)+i\\sin(n\\theta))$ for an integer power $n$.
So if we know the modulus $r$ and the argument $\\theta$ for $z$ then the theorem provides a way of calculating $z^n$.
As usual, you must be careful that the argument is calculated correctly by paying attention to the quadrant of the complex plane in which the complex number lies.
\nAlso remember that for this question, arguments of complex numbers lie in the range $-\\pi \\lt \\theta \\le \\pi$.
\nWith the above in mind we can now answer the questions:
\na)
\n\\[ \\begin{eqnarray*} |\\var{z1}|&=&\\sqrt{(\\var{a1})^2+(\\var{b1})^2}\\\\ &=& \\var{abs(z1)}\\\\ &=&\\var{ans1} \\end{eqnarray*} \\] to 3 decimal places.
\nNote that $r^{\\var{n2}}=|(\\var{z1})^{\\var{n2}}| =\\var{abs(z1)}^{\\var{n2}}=\\var{abs(z1)^n2}$ which we will use in the calculation for $(\\var{z1})^{\\var{n2}}$
\n{m1}.
Hence we see that:
\\[\\begin{eqnarray*} \\arg(\\var{z1}) &=& \\var{arg(z1)}\\\\ &=& \\var{arg1}\\; \\mbox{radians} \\end{eqnarray*} \\] to 3 decimal places.
We have $\\arg((\\var{z1})^{\\var{n2}})=\\var{n2}\\times \\var{arg(z1)} = \\var{n2*arg(z1)}$ radians.
\nHence we have \\[\\begin{eqnarray*}(\\var{z1})^{\\var{n2}} &=& \\var{abs(z1)^n2}(\\cos(\\var{n2*arg(z1)})+\\sin(\\var{n2*arg(z1)})i)\\\\ &=& \\var{abs(z1)^n2}\\cos(\\var{n2*arg(z1)})+\\var{abs(z1)^n2}\\times\\sin(\\var{n2*arg(z1)})i\\\\ &=& \\simplify[std]{{a3}+{b3}i}. \\end{eqnarray*} \\]
\nb)
\n\\[ \\begin{eqnarray*} |\\var{z2}|&=&\\sqrt{(\\var{a2})^2+(\\var{b2})^2}\\\\ &=& \\var{abs(z2)}\\\\ &=&\\var{ans2} \\end{eqnarray*} \\] to 3 decimal places.
\nNote that $r^{\\var{n4}}=|(\\var{z2})^{\\var{n4}}| =\\var{abs(z2)}^{\\var{n4}}=\\var{abs(z2)^n4}$ which we will use in the calculation for $(\\var{z2})^{\\var{n4}}$
\n{m2}.
Hence we see that:
\\[\\begin{eqnarray*} \\arg(\\var{z2}) &=& \\var{arg(z2)}\\\\ &=& \\var{arg2}\\; \\mbox{radians} \\end{eqnarray*} \\] to 3 decimal places.
We have $\\arg((\\var{z2})^{\\var{n4}})=\\var{n4}\\times \\var{arg(z2)} = \\var{n4*arg(z2)}$ radians.
\nHence we have \\[\\begin{eqnarray*}(\\var{z2})^{\\var{n4}} &=& \\var{abs(z2)^n4}(\\cos(\\var{n4*arg(z2)})+\\sin(\\var{n4*arg(z2)})i)\\\\ &=& \\var{abs(z2)^n4}\\cos(\\var{n4*arg(z2)})+\\var{abs(z2)^n4}\\times\\sin(\\var{n4*arg(z2)})i\\\\ &=& \\simplify[std]{{a4}+{b4}i}. \\end{eqnarray*} \\]
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Find the modulus and argument of $\\var{z1}$ correct to 3 decimal places.
\n(i) $|\\var{z1}|\\;=\\;$ [[0]], correct to 3 decimal places.
\n(ii) $\\arg(\\var{z1})\\;=\\;$[[1]] radians, correct to 3 decimal places.
\nHence find:
\n(iii) $(\\var{z1})^{\\var{n2}}\\;=\\;$[[2]]
\nInput as a complex number, with integer values for the real and imaginary parts.
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