$|\\var{z1}|=\\;\\;$[[0]], $\\arg(\\var{z1})=\\;\\;$[[1]] radians
\nInput both answers to 3 decimal places.
\n ", "marks": 0, "gaps": [{"correctAnswerFraction": false, "mustBeReducedPC": 0, "minValue": "ans1-tol", "type": "numberentry", "scripts": {}, "maxValue": "ans1+tol", "variableReplacementStrategy": "originalfirst", "marks": 1, "correctAnswerStyle": "plain", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "allowFractions": false, "variableReplacements": [], "mustBeReduced": false, "showFeedbackIcon": true}, {"correctAnswerFraction": false, "mustBeReducedPC": 0, "minValue": "arg1-tol", "type": "numberentry", "scripts": {}, "maxValue": "arg1+tol", "variableReplacementStrategy": "originalfirst", "marks": 1, "correctAnswerStyle": "plain", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "allowFractions": false, "variableReplacements": [], "mustBeReduced": false, "showFeedbackIcon": true}]}, {"type": "gapfill", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "variableReplacements": [], "scripts": {}, "prompt": "\n$|\\var{z2}|=\\;\\;$[[0]], $\\arg(\\var{z2})=\\;\\;$[[1]] radians
\nInput both answers to 3 decimal places.
\n ", "marks": 0, "gaps": [{"correctAnswerFraction": false, "mustBeReducedPC": 0, "minValue": "ans2-tol", "type": "numberentry", "scripts": {}, "maxValue": "ans2+tol", "variableReplacementStrategy": "originalfirst", "marks": 1, "correctAnswerStyle": "plain", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "allowFractions": false, "variableReplacements": [], "mustBeReduced": false, "showFeedbackIcon": true}, {"correctAnswerFraction": false, "mustBeReducedPC": 0, "minValue": "arg2-tol", "type": "numberentry", "scripts": {}, "maxValue": "arg2+tol", "variableReplacementStrategy": "originalfirst", "marks": 1, "correctAnswerStyle": "plain", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "allowFractions": false, "variableReplacements": [], "mustBeReduced": false, "showFeedbackIcon": true}]}, {"type": "gapfill", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "variableReplacements": [], "scripts": {}, "prompt": "\n$|\\var{z3}|=\\;\\;$[[0]], $\\arg(\\var{z3})=\\;\\;$[[1]] radians
\nInput both answers to 3 decimal places.
\n ", "marks": 0, "gaps": [{"correctAnswerFraction": false, "mustBeReducedPC": 0, "minValue": "ans3-tol", "type": "numberentry", "scripts": {}, "maxValue": "ans3+tol", "variableReplacementStrategy": "originalfirst", "marks": 1, "correctAnswerStyle": "plain", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "allowFractions": false, "variableReplacements": [], "mustBeReduced": false, "showFeedbackIcon": true}, {"correctAnswerFraction": false, "mustBeReducedPC": 0, "minValue": "arg3-tol", "type": "numberentry", "scripts": {}, "maxValue": "arg3+tol", "variableReplacementStrategy": "originalfirst", "marks": 1, "correctAnswerStyle": "plain", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "allowFractions": false, "variableReplacements": [], "mustBeReduced": false, "showFeedbackIcon": true}]}, {"type": "gapfill", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "variableReplacements": [], "scripts": {}, "prompt": "\n$|\\var{z4}|=\\;\\;$[[0]], $\\arg(\\var{z4})=\\;\\;$[[1]] radians
\nInput both answers to 3 decimal places.
\n ", "marks": 0, "gaps": [{"correctAnswerFraction": false, "mustBeReducedPC": 0, "minValue": "ans4-tol", "type": "numberentry", "scripts": {}, "maxValue": "ans4+tol", "variableReplacementStrategy": "originalfirst", "marks": 1, "correctAnswerStyle": "plain", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "allowFractions": false, "variableReplacements": [], "mustBeReduced": false, "showFeedbackIcon": true}, {"correctAnswerFraction": false, "mustBeReducedPC": 0, "minValue": "arg4-tol", "type": "numberentry", "scripts": {}, "maxValue": "arg4+tol", "variableReplacementStrategy": "originalfirst", "marks": 1, "correctAnswerStyle": "plain", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "allowFractions": false, "variableReplacements": [], "mustBeReduced": false, "showFeedbackIcon": true}]}], "tags": [], "variable_groups": [], "statement": "Find the modulus and argument (in radians) of the following complex numbers, where the argument lies between $-\\pi$ and $\\pi$. If your answer is outside this range, simply add or subtract $2\\pi$ to your answer.
\nWhen calculating the argument pay particular attention to the quadrant in which the complex number lies.
\nInput all answers to 3 decimal places.
", "preamble": {"js": "", "css": ""}, "advice": "Note that the arguments $\\theta$ of the complex numbers are in radians and have to be in the range $-\\pi < \\theta \\le \\pi$.
\nYou have to be careful with using a standard calculator when you are finding the argument of a complex number.
\nIf $z=a+bi=r(\\cos(\\theta)+i\\sin(\\theta))$ then we have:$r\\cos(\\theta)=a,\\;\\;r\\sin(\\theta)=b$ and so $\\tan(\\theta) = b/a$.
\nUsing a calculator to find the argument via $\\arctan(b/a)$ works in the range $-\\pi < \\theta \\le \\pi$ when the complex number is in the first or fourth quadrants – you get the correct value.
\nHowever, The calculator gives the wrong value for complex numbers in the other quadrants.
\nSince $\\arctan(b/a)$ does not distinguish between the second and fourth quadrants and the calculator gives the argument for the fourth quadrant you have to add $\\pi$ onto the calculator value.
\nSince $\\arctan(b/a)$ does not distinguish between the first and third quadrants and the calculator gives the argument for the first quadrant you have to take away $\\pi$ from the calculator value.
\n\\[ \\begin{eqnarray*} |\\var{z1}|&=&\\sqrt{(\\var{a1})^2+(\\var{b1})^2}\\\\ &=& \\var{abs(z1)}\\\\ &=&\\var{ans1} \\end{eqnarray*} \\] to 3 decimal places.
\n{m1}
\nHence we see that: \\[\\begin{eqnarray*} \\arg(\\var{z1}) &=& \\var{arg(z1)}\\\\ &=& \\var{arg1}\\; \\mbox{radians} \\end{eqnarray*} \\] to 3 decimal places.
\n\\[ \\begin{eqnarray*} |\\var{z2}|&=&\\sqrt{(\\var{a2})^2+(\\var{b2})^2}\\\\ &=& \\var{abs(z2)}\\\\ &=&\\var{ans2} \\end{eqnarray*} \\] to 3 decimal places.
\n{m2}
\nHence we see that: \\[\\begin{eqnarray*} \\arg(\\var{z2}) &=& \\var{arg(z2)}\\\\ &=& \\var{arg2}\\; \\mbox{radians} \\end{eqnarray*} \\] to 3 decimal places.
\n\\[ \\begin{eqnarray*} |\\var{z3}|&=&\\sqrt{(\\var{c2})^2+(\\var{d2})^2}\\\\ &=& \\var{abs(z3)}\\\\ &=&\\var{ans3} \\end{eqnarray*} \\] to 3 decimal places.
\n{m3}
\nHence we see that: \\[\\begin{eqnarray*} \\arg(\\var{z3}) &=& \\var{arg(z3)}\\\\ &=& \\var{arg3}\\; \\mbox{radians} \\end{eqnarray*} \\] to 3 decimal places.
\n\\[ \\begin{eqnarray*} |\\var{z4}|&=&\\sqrt{(\\var{a3})^2+(\\var{b3})^2}\\\\ &=& \\var{abs(z4)}\\\\ &=&\\var{ans4} \\end{eqnarray*} \\] to 3 decimal places.
\n{m4}
\nHence we see that: \\[\\begin{eqnarray*} \\arg(\\var{z4}) &=& \\var{arg(z4)}\\\\ &=& \\var{arg4}\\; \\mbox{radians} \\end{eqnarray*} \\] to 3 decimal places.
", "metadata": {"description": "Finding the modulus and argument (in radians) of four complex numbers; the arguments between $-\\pi$ and $\\pi$ and careful with quadrants!
", "licence": "Creative Commons Attribution 4.0 International"}, "extensions": [], "type": "question", "contributors": [{"name": "Harry Flynn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/976/"}]}]}], "contributors": [{"name": "Harry Flynn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/976/"}]}