Relations And Functions

  • Question
    CBSEENMA11013101

    Write the complex number space space space space space space space space fraction numerator straight i minus 1 over denominator cos begin display style straight pi over 3 end style plus space straight i space sin begin display style straight pi over 3 end style end fraction in the polar form.


    Solution

    Let    z = i -1 = -1 + i =  r(space cos space straight theta space plus space straight i space sin space straight theta)                                      ....(i)
    rightwards double arrow         r cos space straight theta  = - 1                                                                  ...(ii)
                 r sin <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
#1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493)
#2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs()
#3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...')
#4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array)
#5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array)
#6 {main}</pre>  =   1                                                                  ...(iii)
    Squaring and adding (ii) and (iii) we get
                              r2   = 2                 rightwards double arrow    r = space square root of 2

    From (ii) and (iii).
                   <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
#1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493)
#2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs()
#3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...')
#4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array)
#5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array)
#6 {main}</pre>      rightwards double arrow   space space space space space space space space space cos space straight theta space equals space fraction numerator negative 1 over denominator square root of 2 end fraction
                   space space space space space space space space space square root of 2 space sin space straight theta space equals space 1                rightwards double arrow     space space space space space space space sin space straight theta space equals space fraction numerator 1 over denominator square root of 2 end fraction

    rightwards double arrow                 space space space space space straight theta space equals space 180 degree space minus space 45 degree space equals space 135 degree

     

     space space space space space space space space space space space space space space space space space fraction numerator straight i minus 1 over denominator cos begin display style straight pi over 3 end style plus straight i space sin begin display style straight pi over 3 end style end fraction equals fraction numerator square root of 2 space left parenthesis cos space 135 degree space plus space straight i space sin space 135 degree right parenthesis over denominator cos 60 degree space plus space straight i space sin space 60 degree end fraction


      space space space space space space space space space space space equals square root of 2 equals fraction numerator space left parenthesis cos space 135 degree space plus space straight i space sin space 135 degree right parenthesis open parentheses cos 60 degree minus space straight i space sin 60 degree close parentheses over denominator cos squared 60 degree space plus space space sin squared space 60 degree end fraction

        space space space space space space space space space space space space equals fraction numerator square root of 2 space left parenthesis cos space 135 degree cos 60 degree space plus space sin space 135 degree sin space 60 degree right parenthesis space plus space straight i space left parenthesis cos space 135 degree cos 60 degree space minus space sin space 135 degree sin space 60 degree right parenthesis space over denominator 1 end fraction
       <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
#1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493)
#2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs()
#3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...')
#4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array)
#5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array)
#6 {main}</pre>
          
      space space space space space space space space space space space equals space square root of 2 open square brackets cos 75 degree space plus straight i space sin space 75 degree close square brackets space equals square root of 2 space open square brackets cos open parentheses 75 over 180 straight pi close parentheses plus space straight i space sin open parentheses 75 over 180 cross times straight pi close parentheses close square brackets
      space space space space space space space space space space space space space space space space equals square root of 2 open square brackets cos fraction numerator 5 straight pi over denominator 12 end fraction plus straight i space sin fraction numerator 5 straight pi over denominator 12 end fraction close square brackets
     Which is the  reequired polar form.


             

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