Relations and Functions

Relations and Functions

Question

Convert each of the complex numbers given in the Polar form.
1 - i


Answer

Here, z = 1 - i =
Let z = 1 - i  = space space space space space space space space space space straight r open parentheses cos space straight theta space plus space straight i space sin space straight theta close parentheses                                                                 ..(i)
rightwards double arrow      space space space space space space space space space space space space space space space straight r space cos space straight theta space equals space 1
space space space space space space space space space space space space space space space space straight r space sin space straight theta space equals space minus 1                                                                ..(ii)
                                                                                                        .. (iii)

Squaring and adding (ii) adn (iii), we get 
space space space space space space space space space space space space space space space straight r squared left parenthesis sin squared straight theta space plus space cos squared straight theta right parenthesis space equals space left parenthesis 1 right parenthesis squared plus space left parenthesis negative 1 right parenthesis squared space equals space 1 space plus space 1 space equals space 2
or                         space space space space space space space space space space straight r space equals space square root of 2
From (ii),          space space space space space space space space space space space space space space space cos space straight theta space equals space fraction numerator 1 over denominator square root of 2 end fraction     
From (iii),              space space space space space space space space space sin space straight theta space equals space fraction numerator negative 1 over denominator square root of 2 end fraction
Since, <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> is + ve and space space space sin space straight theta  is - ve

  <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>


Putting in (i), we get space space space space space space space space space space space straight z space equals space square root of 2 open square brackets cos open parentheses fraction numerator negative straight pi over denominator 4 end fraction close parentheses plus straight i space sin space open parentheses fraction numerator negative straight pi over denominator 4 end fraction close parentheses close square brackets
Which is the required polar form.

Hence,        space space space space space space space space space space space space space space space space space space space space space space open vertical bar straight z close vertical bar space space equals space square root of 2 comma space arg space left parenthesis straight z right parenthesis space equals space fraction numerator negative straight pi over denominator 4 end fraction

More Chapters from Relations and Functions