{"id":4820,"date":"2012-11-07T03:46:14","date_gmt":"2012-11-07T03:46:14","guid":{"rendered":"http:\/\/SchoolTutoring.com\/help\/?p=4820"},"modified":"2014-12-02T08:31:57","modified_gmt":"2014-12-02T08:31:57","slug":"polar-representation-of-a-complex-number","status":"publish","type":"post","link":"https:\/\/schooltutoring.com\/help\/polar-representation-of-a-complex-number\/","title":{"rendered":"Polar Representation of a Complex Number"},"content":{"rendered":"

In polar representation, a complex number is represented in terms of two variables r <\/em><\/strong>and t<\/em><\/strong>, where r is the modulus of complex number and t is the angle with the positive direction of x-axis.<\/p>\n

We know that the modulus of a complex number z=a+ib is denoted by |z| and is defined as \"\"<\/a>. We also know that the point P(a,b) is the representation of the complex number a+bi in a coordinate plane.<\/p>\n

\"\"<\/a><\/p>\n

Then from the above figure,<\/p>\n

Cos t = a\/r ==> a = r Cost<\/p>\n

Sin t = b\/r ==>b= r Sint,<\/p>\n

where t<\/em><\/strong> is the angle in radians which is called the argument of a+bi.<\/p>\n

and we have tant = b\/a==> t = tan-1<\/sup>(b\/a)<\/p>\n

Aslo, from trigonometric Pythagorean identity, we have<\/p>\n

Cos2<\/sup>t +Sin2<\/sup>t = 1<\/p>\n

a2<\/sup>\/r2<\/sup> + b2<\/sup>\/r2<\/sup> = 1<\/p>\n

r2<\/sup> = a2<\/sup>+b2<\/sup><\/p>\n

r = \u221a(a2<\/sup>+b2<\/sup>). So, r<\/em><\/strong> is nothing but the modulus of a+bi.<\/p>\n

Thus, we can write a complex number a+bi as follows.<\/p>\n

a+bi= r(Cost + i Sin t),<\/em><\/strong> where<\/p>\n

r= \u221a a2<\/sup>+b2<\/sup><\/p>\n

and t = tan-1<\/sup>(b\/a).<\/p>\n

The above representation is called polar form<\/em><\/strong> of a+ib.<\/p>\n

Converting a given complex number into polar form:<\/h4>\n

(1)\u00a0\u00a0\u00a0 Find r = \u221a a2<\/sup>+b2<\/sup><\/p>\n

(2)\u00a0\u00a0\u00a0 Find t = tan-1<\/sup>(b\/a).<\/p>\n

(3)\u00a0\u00a0\u00a0 Substitute r and t in the polar form r(Cos t + i Sint)<\/p>\n

Example:<\/strong><\/p>\n

1)\u00a0\u00a0\u00a0\u00a0\u00a0 i = 1 (Cos \u03c0\/2 + i Sin \u03c0\/2).<\/p>\n

2)\u00a0\u00a0\u00a0\u00a0\u00a0 1+I<\/p>\n

R = \u221a a2<\/sup>+b2<\/sup>=\u221a2<\/p>\n

t = tan-1<\/sup>(b\/a)=\u00a0 tan-1<\/sup>(1)=\u03c0\/4.<\/p>\n

So, 1+I = \u221a2 (Cos \u03c0\/4 + i Sin \u03c0\/4)<\/p>\n

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SchoolTutoring Academy<\/span> is the premier educational services company for K-12 and college students. We offer tutoring programs for students in K-12, AP classes, and college. To learn more about how we help parents and students in Manitoba visit: Tutoring in Manitoba. <\/a><\/p>\n

Example image courtesy of\u00a0 https:\/\/www.ping.be\/~ping1339\/complget.htm<\/p>\n","protected":false},"excerpt":{"rendered":"

In polar representation, a complex number is represented in terms of two variables r and t, where r is the modulus of complex number and t is the angle with the positive direction of x-axis. We know that the modulus of a complex number z=a+ib is denoted by |z| and is defined as . We […]<\/p>\n","protected":false},"author":19,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"inline_featured_image":false,"footnotes":""},"categories":[2],"tags":[94,135,397,677,1137,1350],"class_list":["post-4820","post","type-post","status-publish","format-standard","hentry","category-algebra","tag-angle","tag-argument","tag-conversion","tag-form","tag-modulus","tag-polar"],"acf":[],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/posts\/4820","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/users\/19"}],"replies":[{"embeddable":true,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/comments?post=4820"}],"version-history":[{"count":0,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/posts\/4820\/revisions"}],"wp:attachment":[{"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/media?parent=4820"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/categories?post=4820"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/tags?post=4820"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}