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If your answer is outside this range, simply add or subtract $2\\pi$ to your answer.</p>\n<p>When calculating the argument pay particular attention to the quadrant in which the complex number lies.</p>\n<p>Input all answers to 3 decimal places.</p>\n<p>*Hint - draw an Argand diagram to help!*</p>", "advice": "<p>Note that the arguments $\\theta$ of the complex numbers are in radians and have to be in the range $-\\pi &lt; \\theta \\le \\pi$.</p>\n<p>You have to be careful with using a standard calculator when you are finding the argument of a complex number.</p>\n<p>If $z=a+bi=r(\\cos(\\theta)+i\\sin(\\theta))$ then we have:$r\\cos(\\theta)=a,\\;\\;r\\sin(\\theta)=b$ and so $\\tan(\\theta) = b/a$.</p>\n<p>Using a calculator to find the argument via $\\arctan(b/a)$ works in the range $-\\pi &lt; \\theta \\le \\pi$ when the complex number is in the first or fourth quadrants &ndash; you get the correct value.</p>\n<p>However, The calculator gives the wrong value for complex numbers in the other quadrants.</p>\n<h4>Complex number in the Second Quadrant.</h4>\n<p>Since $\\arctan(b/a)$ 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.</p>\n<h4>Complex number in the Third Quadrant.</h4>\n<p>Since $\\arctan(b/a)$ 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.</p>\n<h4>a)</h4>\n<h5>Modulus</h5>\n<p>\\[ \\begin{eqnarray*} |\\var{z1}|&amp;=&amp;\\sqrt{(\\var{a1})^2+(\\var{b1})^2}\\\\ &amp;=&amp; \\var{abs(z1)}\\\\ &amp;=&amp;\\var{ans1} \\end{eqnarray*} \\] to 3 decimal places.</p>\n<h5>Argument</h5>\n<p><strong>{m1}</strong></p>\n<p>Hence we see that: \\[\\begin{eqnarray*} \\arg(\\var{z1}) &amp;=&amp; \\var{arg(z1)}\\\\ &amp;=&amp; \\var{arg1}\\; \\mbox{radians} \\end{eqnarray*} \\] to 3 decimal places.</p>\n<h4>b)</h4>\n<h5>Modulus</h5>\n<p>\\[ \\begin{eqnarray*} |\\var{z2}|&amp;=&amp;\\sqrt{(\\var{a2})^2+(\\var{b2})^2}\\\\ &amp;=&amp; \\var{abs(z2)}\\\\ &amp;=&amp;\\var{ans2} \\end{eqnarray*} \\] to 3 decimal places.</p>\n<h5>Argument</h5>\n<p><strong>{m2}</strong></p>\n<p>Hence we see that: \\[\\begin{eqnarray*} \\arg(\\var{z2}) &amp;=&amp; \\var{arg(z2)}\\\\ &amp;=&amp; \\var{arg2}\\; \\mbox{radians} \\end{eqnarray*} \\] to 3 decimal places.</p>\n<h4>c)</h4>\n<h5>Modulus</h5>\n<p>\\[ \\begin{eqnarray*} |\\var{z3}|&amp;=&amp;\\sqrt{(\\var{c2})^2+(\\var{d2})^2}\\\\ &amp;=&amp; \\var{abs(z3)}\\\\ &amp;=&amp;\\var{ans3} \\end{eqnarray*} \\] to 3 decimal places.</p>\n<h5>Argument</h5>\n<p><strong>{m3}</strong></p>\n<p>Hence we see that: \\[\\begin{eqnarray*} \\arg(\\var{z3}) &amp;=&amp; \\var{arg(z3)}\\\\ &amp;=&amp; \\var{arg3}\\; \\mbox{radians} \\end{eqnarray*} \\] to 3 decimal places.</p>\n<h4>d)</h4>\n<h5>Modulus</h5>\n<p>\\[ \\begin{eqnarray*} 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