Principle of Mathematical Induction

Principle of Mathematical Induction

Question

Find the maximum and minimum values of:

straight a space cosx plus straight b space sinx

Answer

Let            straight u space equals space acosx plus bsinx
Put straight a equals straight r space cosβ comma space space space straight b equals straight r space sinβ
Squaring and adding, we get
                          straight a squared plus straight b squared space equals space straight r to the power of 2 space end exponent left parenthesis cos squared straight beta plus sin squared straight beta right parenthesis
         rightwards double arrow                      straight a squared plus straight b squared space equals space straight r squared
rightwards double arrow                          straight r equals square root of straight a squared plus straight b squared end root
rightwards double arrow                          straight u equals space straight r space cos space straight beta space cosx space plus space straight r space sin space straight beta space space sinx space equals space straight r space left square bracket cosβ space cosx space plus space sinβ space sinx right square bracket
∴                         space space space space straight u space equals space square root of straight a squared plus straight b squared space end root cos left parenthesis straight beta minus straight x right parenthesis
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#6 {main}</pre>     (∵ cos x and sin x lie between -1 and 1)
rightwards double arrow           space space space minus 1 space cross times space square root of straight a squared plus straight b squared end root less or equal than square root of straight a squared plus straight b squared end root space cos left parenthesis straight beta minus straight x right parenthesis space less or equal than space 1 space cross times space square root of straight a squared plus straight b squared end root  
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#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 minus square root of straight a squared plus straight b squared end root space less or equal than space straight a space cosx space plus straight b space sinx less or equal than square root of straight a squared plus straight b squared end root        
Hence,         max. value of {a cosx + b sinx} = square root of straight a squared plus straight b squared end root
                  Min. value of {a cosx + b sinx} = - square root of straight a squared plus straight b squared end root
 
                   

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