{"id":2280,"date":"2025-08-12T17:52:46","date_gmt":"2025-08-12T17:52:46","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=2280"},"modified":"2025-08-13T17:04:43","modified_gmt":"2025-08-13T17:04:43","slug":"polar-form-of-complex-numbers-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/polar-form-of-complex-numbers-learn-it-2\/","title":{"raw":"Polar Form of Complex Numbers: Learn It 2","rendered":"Polar Form of Complex Numbers: Learn It 2"},"content":{"raw":"

Writing Complex Numbers in Polar Form<\/h2>\r\nThe polar form of a complex number<\/strong> expresses a number in terms of an angle [latex]\\theta [\/latex] and its distance from the origin [latex]r[\/latex]. Given a complex number in rectangular form<\/strong> expressed as [latex]z=x+yi[\/latex], we use the same conversion formulas as we do to write the number in trigonometric form:\r\n
[latex]\\begin{gathered}x=r\\cos \\theta \\\\ y=r\\sin \\theta \\\\ r=\\sqrt{{x}^{2}+{y}^{2}} \\end{gathered}[\/latex]<\/div>\r\n\"Triangle\r\n\r\nWe use the term modulus<\/strong> to represent the absolute value of a complex number, or the distance from the origin to the point [latex]\\left(x,y\\right)[\/latex]. The modulus, then, is the same as [latex]r[\/latex], the radius in polar form. We use [latex]\\theta [\/latex] to indicate the angle of direction (just as with polar coordinates). Substituting, we have\r\n
[latex]\\begin{align}&z=x+yi \\\\ &z=r\\cos \\theta +\\left(r\\sin \\theta \\right)i \\\\ &z=r\\left(\\cos \\theta +i\\sin \\theta \\right) \\end{align}[\/latex]<\/div>\r\n
\r\n
\r\n

modulus<\/h3>\r\nThe distance from the origin to the point represented by the complex number. This can be found as the absolute value of the complex number.\r\n\r\n<\/div>\r\n<\/section>
\r\n

polar form of a complex number<\/h3>\r\nWriting a complex number in polar form involves the following conversion formulas:\r\n

[latex]\\begin{gathered} x=r\\cos \\theta \\\\ y=r\\sin \\theta \\\\ r=\\sqrt{{x}^{2}+{y}^{2}} \\end{gathered}[\/latex]<\/p>\r\nMaking a direct substitution, we have\r\n

[latex]\\begin{align}&z=x+yi \\\\ &z=\\left(r\\cos \\theta \\right)+i\\left(r\\sin \\theta \\right) \\\\ &z=r\\left(\\cos \\theta +i\\sin \\theta \\right) \\end{align}[\/latex]<\/p>\r\nwhere [latex]r[\/latex] is the modulus<\/strong> and [latex]\\theta [\/latex] is the argument<\/strong>. We often use the abbreviation [latex]r\\text{cis}\\theta [\/latex] to represent [latex]r\\left(\\cos \\theta +i\\sin \\theta \\right)[\/latex].\r\n\r\n<\/section><\/div>\r\n

Express the complex number [latex]4i[\/latex] using polar coordinates.[reveal-answer q=\"3488\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"3488\"]\r\n\r\nOn the complex plane, the number [latex]z=4i[\/latex] is the same as [latex]z=0+4i[\/latex]. Writing it in polar form, we have to calculate [latex]r[\/latex] first.\r\n

[latex]\\begin{align}&r=\\sqrt{{x}^{2}+{y}^{2}} \\\\ &r=\\sqrt{{0}^{2}+{4}^{2}} \\\\ &r=\\sqrt{16} \\\\ &r=4 \\end{align}[\/latex]<\/p>\r\nNext, we look at [latex]x[\/latex]. If [latex]x=r\\cos \\theta [\/latex], and [latex]x=0[\/latex], then [latex]\\theta =\\frac{\\pi }{2}[\/latex]. In polar coordinates, the complex number [latex]z=0+4i[\/latex] can be written as [latex]z=4\\left(\\cos \\left(\\frac{\\pi }{2}\\right)+i\\sin \\left(\\frac{\\pi }{2}\\right)\\right)[\/latex] or [latex]4\\text{cis}\\left(\\frac{\\pi }{2}\\right)[\/latex].\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Plot Figure 6<\/b>[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>

\r\n
\r\n\r\nExpress [latex]z=3i[\/latex] as [latex]r\\text{cis}\\theta [\/latex] in polar form.\r\n\r\n[reveal-answer q=\"364839\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"364839\"]\r\n\r\n[latex]z=3\\left(\\cos \\left(\\frac{\\pi }{2}\\right)+i\\sin \\left(\\frac{\\pi }{2}\\right)\\right)[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/section>
Find the polar form of [latex]-4+4i[\/latex].[reveal-answer q=\"871564\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"871564\"]\r\n\r\nFirst, find the value of [latex]r[\/latex].\r\n

[latex]\\begin{align}&r=\\sqrt{{x}^{2}+{y}^{2}} \\\\ &r=\\sqrt{{\\left(-4\\right)}^{2}+\\left({4}^{2}\\right)} \\\\ &r=\\sqrt{32} \\\\ &r=4\\sqrt{2} \\end{align}[\/latex]<\/p>\r\nFind the angle [latex]\\theta [\/latex] using the formula:\r\n

[latex]\\begin{align}&\\cos \\theta =\\frac{x}{r} \\\\ &\\cos \\theta =\\frac{-4}{4\\sqrt{2}} \\\\ &\\cos \\theta =-\\frac{1}{\\sqrt{2}} \\\\ &\\theta ={\\cos }^{-1}\\left(-\\frac{1}{\\sqrt{2}}\\right)=\\frac{3\\pi }{4} \\end{align}[\/latex]<\/p>\r\nThus, the solution is [latex]4\\sqrt{2}\\cos\\left(\\frac{3\\pi }{4}\\right)[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>

Write [latex]z=\\sqrt{3}+i[\/latex] in polar form.[reveal-answer q=\"164997\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"164997\"][latex]z=2\\left(\\cos \\left(\\frac{\\pi }{6}\\right)+i\\sin \\left(\\frac{\\pi }{6}\\right)\\right)[\/latex][\/hidden-answer]\r\n\r\n<\/section>
[ohm_question hide_question_numbers=1]173845[\/ohm_question]<\/section>","rendered":"

Writing Complex Numbers in Polar Form<\/h2>\n

The polar form of a complex number<\/strong> expresses a number in terms of an angle [latex]\\theta[\/latex] and its distance from the origin [latex]r[\/latex]. Given a complex number in rectangular form<\/strong> expressed as [latex]z=x+yi[\/latex], we use the same conversion formulas as we do to write the number in trigonometric form:<\/p>\n

[latex]\\begin{gathered}x=r\\cos \\theta \\\\ y=r\\sin \\theta \\\\ r=\\sqrt{{x}^{2}+{y}^{2}} \\end{gathered}[\/latex]<\/div>\n

\"Triangle<\/p>\n

We use the term modulus<\/strong> to represent the absolute value of a complex number, or the distance from the origin to the point [latex]\\left(x,y\\right)[\/latex]. The modulus, then, is the same as [latex]r[\/latex], the radius in polar form. We use [latex]\\theta[\/latex] to indicate the angle of direction (just as with polar coordinates). Substituting, we have<\/p>\n

[latex]\\begin{align}&z=x+yi \\\\ &z=r\\cos \\theta +\\left(r\\sin \\theta \\right)i \\\\ &z=r\\left(\\cos \\theta +i\\sin \\theta \\right) \\end{align}[\/latex]<\/div>\n
\n
\n
\n

modulus<\/h3>\n

The distance from the origin to the point represented by the complex number. This can be found as the absolute value of the complex number.<\/p>\n<\/div>\n<\/section>\n

\n

polar form of a complex number<\/h3>\n

Writing a complex number in polar form involves the following conversion formulas:<\/p>\n

[latex]\\begin{gathered} x=r\\cos \\theta \\\\ y=r\\sin \\theta \\\\ r=\\sqrt{{x}^{2}+{y}^{2}} \\end{gathered}[\/latex]<\/p>\n

Making a direct substitution, we have<\/p>\n

[latex]\\begin{align}&z=x+yi \\\\ &z=\\left(r\\cos \\theta \\right)+i\\left(r\\sin \\theta \\right) \\\\ &z=r\\left(\\cos \\theta +i\\sin \\theta \\right) \\end{align}[\/latex]<\/p>\n

where [latex]r[\/latex] is the modulus<\/strong> and [latex]\\theta[\/latex] is the argument<\/strong>. We often use the abbreviation [latex]r\\text{cis}\\theta[\/latex] to represent [latex]r\\left(\\cos \\theta +i\\sin \\theta \\right)[\/latex].<\/p>\n<\/section>\n<\/div>\n

Express the complex number [latex]4i[\/latex] using polar coordinates.<\/p>\n