Principle of Mathematical Induction

Principle of Mathematical Induction

Question

Angles A, B, C of a triangle ABC are in A.P. and straight b colon straight c space equals space square root of 3 colon square root of 2. Find the angles of the triangle.

Answer

A, B, C are in A.P.
rightwards double arrow               2B = A + C                                                         ...(i)
Also,           straight A plus straight B plus straight C space equals space 180 degree              (∵ The sum of the angles of a straight capital delta space equals space 180 degree)
rightwards double arrow              straight B space plus space left parenthesis 2 straight B right parenthesis space equals space 180 degree
rightwards double arrow                     3 straight B space equals space 180 degree
∴                          angle straight B space equals space 60 degree                                                ...(ii)
Now,            straight b colon straight c space equals space square root of 3 colon square root of 2
Also,            fraction numerator straight a over denominator sin space straight A end fraction space equals space fraction numerator straight b over denominator sin space straight B end fraction space equals space fraction numerator straight c over denominator sin space straight C end fraction space equals straight k                              (By sine formula)

rightwards double arrow                    space space space straight b over straight c space equals space sinB over sinC             <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)
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#6 {main}</pre>

rightwards double arrow                     sin space straight C space equals space fraction numerator square root of 2 space sinB over denominator square root of 3 end fraction space equals space fraction numerator square root of 2 over denominator square root of 3 end fraction sin space 60 degree                      (∴ angle straight B space equals space 60 degree)
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)
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#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                         angle straight C space equals space 45 degree
Now,             straight A plus straight B plus straight C space equals space 180 degree
rightwards double arrow           straight A plus 60 degree plus 45 degree space equals space 180 degree
or                             straight A equals space 180 degree minus 105 degree space equals space 75 degree 
Hence, the angles of the triangle are:
angle straight A space equals space 75 degree space equals space fraction numerator 5 straight pi over denominator 12 end fraction semicolon space angle straight B space equals space 60 degree space equals space straight pi over 3 semicolon space space angle straight C space equals space 45 degree space equals space straight pi over 4

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