Principle of Mathematical Induction

Principle of Mathematical Induction

Question

Find the general solution of:

space space space 4 space sin squared straight theta space minus space 8 space cosθ space plus space 1 space equals 0

Answer
space space space space space space space 4 space sin squared straight theta space minus space 8 space cosθ space plus space 1 space equals space 0                        rightwards double arrow  4 space left parenthesis 1 minus cos squared straight theta right parenthesis space minus space 8 space cosθ space plus space 1 space equals space 0

rightwards double arrow  <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   4 space cos squared straight theta space plus space 10 space cosθ space minus space 2 space cosθ space minus space 5 space equals space 0

rightwards double arrow   2 space cosθ space left parenthesis 2 space cosθ space plus space 5 right parenthesis space minus space 1 left parenthesis 2 space cosθ space plus space 5 right parenthesis space equals space 0 space space space rightwards double arrow space space space space space left parenthesis 2 space cosθ space minus space 1 right parenthesis space left parenthesis 2 space cosθ space plus space 5 right parenthesis space equals space 0

rightwards double arrow      either   2 space cosθ space minus space 1 space equals space 0                          or    cos space straight theta space equals space minus 5 over 2

rightwards double arrow                  cos space straight theta space equals space 1 half                                                               [∵ negative 1 less or equal than cosθ less or equal than 1]
rightwards double arrow                    cos space straight theta space equals space cos space straight pi over 3
                     rightwards double arrow          straight theta space equals space 2 nπ space plus-or-minus space straight pi over 3 comma space space straight n space element of space straight Z              

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