{"id":2281,"date":"2025-08-12T17:52:48","date_gmt":"2025-08-12T17:52:48","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=2281"},"modified":"2025-08-13T17:04:59","modified_gmt":"2025-08-13T17:04:59","slug":"polar-form-of-complex-numbers-learn-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/polar-form-of-complex-numbers-learn-it-3\/","title":{"raw":"Polar Form of Complex Numbers: Learn It 3","rendered":"Polar Form of Complex Numbers: Learn It 3"},"content":{"raw":"

Converting a Complex Number from Polar to Rectangular Form<\/h2>\r\nConverting a complex number from polar form to rectangular form is a matter of evaluating what is given and using the distributive property. In other words, given [latex]z=r\\left(\\cos \\theta +i\\sin \\theta \\right)[\/latex], first evaluate the trigonometric functions [latex]\\cos \\theta [\/latex] and [latex]\\sin \\theta [\/latex]. Then, multiply through by [latex]r[\/latex].\r\n\r\n
Convert the polar form of the given complex number to rectangular form:\r\n

[latex]z=12\\left(\\cos \\left(\\frac{\\pi }{6}\\right)+i\\sin \\left(\\frac{\\pi }{6}\\right)\\right)[\/latex]<\/p>\r\n[reveal-answer q=\"145161\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"145161\"]\r\n\r\nWe begin by evaluating the trigonometric expressions.\r\n

[latex]\\begin{gathered}\\cos \\left(\\frac{\\pi }{6}\\right)=\\frac{\\sqrt{3}}{2}\\\\\\sin \\left(\\frac{\\pi }{6}\\right)=\\frac{1}{2}\\end{gathered}[\/latex]<\/p>\r\nAfter substitution, the complex number is\r\n

[latex]z=12\\left(\\frac{\\sqrt{3}}{2}+\\frac{1}{2}i\\right)[\/latex]<\/p>\r\nWe apply the distributive property:\r\n

[latex]\\begin{align}z&=12\\left(\\frac{\\sqrt{3}}{2}+\\frac{1}{2}i\\right) \\\\ &=\\left(12\\right)\\frac{\\sqrt{3}}{2}+\\left(12\\right)\\frac{1}{2}i \\\\ &=6\\sqrt{3}+6i \\end{align}[\/latex]<\/p>\r\nThe rectangular form of the given point in complex form is [latex]6\\sqrt{3}+6i[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>

Find the rectangular form of the complex number given [latex]r=13[\/latex] and [latex]\\tan \\theta =\\frac{5}{12}[\/latex].[reveal-answer q=\"11326\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"11326\"]\r\n\r\nIf [latex]\\tan \\theta =\\frac{5}{12}[\/latex], and [latex]\\tan \\theta =\\frac{y}{x}[\/latex], we first determine [latex]r=\\sqrt{{x}^{2}+{y}^{2}}=\\sqrt{{12}^{2}+{5}^{2}}=13\\text{.}[\/latex] We then find [latex]\\cos \\theta =\\frac{x}{r}[\/latex] and [latex]\\sin \\theta =\\frac{y}{r}[\/latex].\r\n

[latex]\\begin{align}z&=13\\left(\\cos \\theta +i\\sin \\theta \\right) \\\\ &=13\\left(\\frac{12}{13}+\\frac{5}{13}i\\right) \\\\ &=12+5i \\end{align}[\/latex]<\/p>\r\nThe rectangular form of the given number in complex form is [latex]12+5i[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>

Convert the complex number to rectangular form:\r\n

[latex]z=4\\left(\\cos \\frac{11\\pi }{6}+i\\sin \\frac{11\\pi }{6}\\right)[\/latex]<\/p>\r\n

[reveal-answer q=\"9609\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"9609\"]<\/p>\r\n

[latex]z=2\\sqrt{3}-2i[\/latex]<\/p>\r\n

[\/hidden-answer]<\/p>\r\n\r\n<\/section>

[ohm_question hide_question_numbers=1]173840[\/ohm_question]<\/section>","rendered":"

Converting a Complex Number from Polar to Rectangular Form<\/h2>\n

Converting a complex number from polar form to rectangular form is a matter of evaluating what is given and using the distributive property. In other words, given [latex]z=r\\left(\\cos \\theta +i\\sin \\theta \\right)[\/latex], first evaluate the trigonometric functions [latex]\\cos \\theta[\/latex] and [latex]\\sin \\theta[\/latex]. Then, multiply through by [latex]r[\/latex].<\/p>\n

Convert the polar form of the given complex number to rectangular form:<\/p>\n

[latex]z=12\\left(\\cos \\left(\\frac{\\pi }{6}\\right)+i\\sin \\left(\\frac{\\pi }{6}\\right)\\right)[\/latex]<\/p>\n