{"id":35176,"date":"2022-08-25T11:06:12","date_gmt":"2022-08-25T05:36:12","guid":{"rendered":"https:\/\/apboardsolutions.guru\/?p=35176"},"modified":"2022-08-25T11:06:12","modified_gmt":"2022-08-25T05:36:12","slug":"inter-2nd-year-maths-2a-de-moivres-theorem-formulas","status":"publish","type":"post","link":"https:\/\/apboardsolutions.guru\/inter-2nd-year-maths-2a-de-moivres-theorem-formulas\/","title":{"rendered":"Inter 2nd Year Maths 2A De Moivre\u2019s Theorem Formulas"},"content":{"rendered":"<p>Use these <a href=\"https:\/\/apboardsolutions.guru\/inter-2nd-year-maths-2a-formulas\/\">Inter 2nd Year Maths 2A Formulas<\/a> PDF Chapter 2 De Moivre\u2019s Theorem to solve questions creatively.<\/p>\n<h2>Intermediate 2nd Year Maths 2A De Moivre\u2019s Theorem Formulas<\/h2>\n<p><strong>Statement:<\/strong><br \/>\n\u2192 If \u2018n\u2019 is an integer, then (cos \u03b8 + i sin \u03b8)<sup>n<\/sup> = cos n\u03b8 + i sin n\u03b8<br \/>\nIf n\u2019 is a rational number, then one of the values of<br \/>\n(cos \u03b8 + i sin \u03b8)<sup>n<\/sup> is cos n\u03b8 + i sin n\u03b8<\/p>\n<p><strong>n<sup>th<\/sup> roots of unity:<\/strong><br \/>\n\u2192 n<sup>th<\/sup> roots of unity are {1, \u03c9, \u03c9<sup>2<\/sup> &#8230;&#8230;.. \u03c9<sup>n &#8211; 1<\/sup>}.<br \/>\nWhere \u03c9 = \\(\\left[\\cos \\frac{2 k \\pi}{n}+i \\sin \\frac{2 k \\pi}{n}\\right]\\) k = 0, 1, 2 &#8230;&#8230;. (n &#8211; 1).<br \/>\nIf \u03c9 is a n<sup>th<\/sup> root of unity, then<\/p>\n<ul>\n<li>\u03c9<sup>n<\/sup> = 1<\/li>\n<li>1 + \u03c9 + \u03c9<sup>2<\/sup> + &#8230;&#8230;&#8230;&#8230; + \u03c9<sup>n &#8211; 1<\/sup> = 0<\/li>\n<\/ul>\n<p><img loading=\"lazy\" src=\"https:\/\/apboardsolutions.com\/wp-content\/uploads\/2022\/07\/AP-Board-Solutions.png\" alt=\"Inter 2nd Year Maths 2A De Moivre\u2019s Theorem Formulas\" width=\"196\" height=\"17\" \/><\/p>\n<p><strong>Cube roots of unity:<\/strong><br \/>\n\u2192 1, \u03c9, \u03c9<sup>2<\/sup> are cube roots of unity when<\/p>\n<ul>\n<li>\u03c9<sup>3<\/sup> = 1<\/li>\n<li>1 + \u03c9 + \u03c9<sup>2<\/sup> = 0<\/li>\n<li>\u03c9 = \\(\\frac{-1+i \\sqrt{3}}{2}\\), \u03c9<sup>2<\/sup> = \\(\\frac{-1-i \\sqrt{3}}{2}\\)<\/li>\n<li>Fourth roots of unity roots are 1, &#8211; 1, i, &#8211; i<\/li>\n<\/ul>\n<p>\u2192 If Z<sub>0<\/sub> = r<sub>0<\/sub> cis \u03b8<sub>0<\/sub> \u2260 0, then the n<sup>th<\/sup> roots of Z<sub>0<\/sub> are \u03b1<sub>k<\/sub> = (r<sub>0<\/sub>)<sup>1\/n<\/sup> cis\\(\\left(\\frac{2 k \\pi+\\theta_{0}}{n}\\right)\\) where k = 0, 1, 2, &#8230;&#8230;&#8230; (n &#8211; 1)<\/p>\n<p>\u2192 If n is any integer, (cos \u03b8 + i sin \u03b8)<sup>n<\/sup> = cos n\u03b8 + i sin n\u03b8<\/p>\n<p>\u2192 If n is any fraction, one of the values of (cos\u03b8 + i sin\u03b8)<sup>n<\/sup> is cos n\u03b8 + i sin n\u03b8.<\/p>\n<p>\u2192 (sin\u03b8 + i cos\u03b8)<sup>n<\/sup> = cos(\\(\\frac{n \\pi}{2}\\) &#8211; n\u03b8) + i sin(\\(\\frac{n \\pi}{2}\\) &#8211; n\u03b8)<\/p>\n<p>\u2192 If x = cos\u03b8 + i sin\u03b8, then x + \\(\\frac{1}{x}\\) = 2 cos\u03b8, x &#8211; \\(\\frac{1}{x}\\) = 2i sin\u03b8<\/p>\n<p>\u2192 x<sup>n<\/sup> + \\(\\frac{1}{x^{n}}\\) = 2cos n\u03b8, x<sup>n<\/sup> &#8211; \\(\\frac{1}{x^{n}}\\) = 2i sin n\u03b8<\/p>\n<p>\u2192 The nth roots of a complex number form a G.P. with common ratio cis\\(\\frac{2 \\pi}{n}\\) which is denoted by \u03c9.<\/p>\n<p>\u2192 The points representing n<sup>th<\/sup> roots of a complex number in the Argand diagram are concyclic.<\/p>\n<p>\u2192 The points representing n<sup>th<\/sup> roots of a complex number in the Argand diagram form a regular polygon of n sides.<\/p>\n<p>\u2192 The points representing the cube roots of a complex number in the Argand diagram form an equilateral triangle.<\/p>\n<p>\u2192 The points representing the fourth roots of complex number in the Argand diagram form a square.<\/p>\n<p>\u2192 The nth roots of unity are 1, w, w<sup>2<\/sup>,&#8230;&#8230;&#8230;. , w<sup>n-1<\/sup> where w = cis\\(\\frac{2 \\pi}{n}\\)<\/p>\n<p><img loading=\"lazy\" src=\"https:\/\/apboardsolutions.com\/wp-content\/uploads\/2022\/07\/AP-Board-Solutions.png\" alt=\"Inter 2nd Year Maths 2A De Moivre\u2019s Theorem Formulas\" width=\"196\" height=\"17\" \/><\/p>\n<p>\u2192 The sum of the nth roots of unity is zero (or) the sum of the nth roots of any complex number is zero.<\/p>\n<p>\u2192 The cube roots of unity are 1, \u03c9, \u03c9<sup>2<\/sup> where \u03c9 = cis\\(\\frac{2 \\pi}{3}\\), \u03c9<sup>2<\/sup> = cis\\(\\frac{4 \\pi}{3}\\) or<br \/>\n\u03c9 = \\(\\frac{-1+i \\sqrt{3}}{2}\\)<br \/>\n\u03c9<sup>2<\/sup> = \\(\\frac{-1-i \\sqrt{3}}{2}\\)<br \/>\n1 + \u03c9 + \u03c9<sup>2<\/sup> = 0<br \/>\n\u03c9<sup>3<\/sup> = 1<\/p>\n<p>\u2192 The product of the n<sup>th<\/sup> roots of unity is (-1)<sup>n-1<\/sup> .<\/p>\n<p>\u2192 The product of the n<sup>th<\/sup> roots of a complex number Z is Z(-1)<sup>n-1<\/sup> .<\/p>\n<p>\u2192 \u03c9, \u03c9<sup>2<\/sup> are the roots of the equation x<sup>2<\/sup> + x + 1 = 0<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Use these Inter 2nd Year Maths 2A Formulas PDF Chapter 2 De Moivre\u2019s Theorem to solve questions creatively. Intermediate 2nd Year Maths 2A De Moivre\u2019s Theorem Formulas Statement: \u2192 If \u2018n\u2019 is an integer, then (cos \u03b8 + i sin \u03b8)n = cos n\u03b8 + i sin n\u03b8 If n\u2019 is a rational number, then &hellip;<\/p>\n","protected":false},"author":5,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":""},"categories":[14],"tags":[],"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/apboardsolutions.guru\/wp-json\/wp\/v2\/posts\/35176"}],"collection":[{"href":"https:\/\/apboardsolutions.guru\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/apboardsolutions.guru\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/apboardsolutions.guru\/wp-json\/wp\/v2\/users\/5"}],"replies":[{"embeddable":true,"href":"https:\/\/apboardsolutions.guru\/wp-json\/wp\/v2\/comments?post=35176"}],"version-history":[{"count":1,"href":"https:\/\/apboardsolutions.guru\/wp-json\/wp\/v2\/posts\/35176\/revisions"}],"predecessor-version":[{"id":35177,"href":"https:\/\/apboardsolutions.guru\/wp-json\/wp\/v2\/posts\/35176\/revisions\/35177"}],"wp:attachment":[{"href":"https:\/\/apboardsolutions.guru\/wp-json\/wp\/v2\/media?parent=35176"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/apboardsolutions.guru\/wp-json\/wp\/v2\/categories?post=35176"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/apboardsolutions.guru\/wp-json\/wp\/v2\/tags?post=35176"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}