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Trigonometric Functions

Short Answer Type

51. In any quadrilateral ABCD, show that sin (A + B) + sin (C + D) = 0.

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52. If A, B, C, D are the vertices of a cyclic quadrilateral, show that sin A + sin B - sin C - sin D = 0.

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53.

Show that : $<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$

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54. Find the period of the following trigonometric functions:

sin open parentheses fraction numerator 3 straight x over denominator 2 end fraction close parentheses

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55. Find the period of the following trigonometric functions:

tan open parentheses fraction numerator 5 straight x over denominator 3 end fraction close parentheses

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56.

Show that: $cos open parentheses fraction numerator 3 straight pi over denominator 2 end fraction plus straight x close parentheses space cos left parenthesis 2 straight pi plus straight x right parenthesis space open square brackets cot open parentheses fraction numerator 3 straight pi over denominator 2 end fraction minus straight x close parentheses plus cot left parenthesis 2 straight pi plus straight x right parenthesis close square brackets space equals space 1$

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57. Find the value of:

sin open parentheses straight pi over 12 close parentheses

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58. Find the value of:

$space space space cos open parentheses fraction numerator 5 straight pi over denominator 12 end fraction close parentheses$

space space space cos open parentheses fraction numerator 5 straight pi over denominator 12 end fraction close parentheses space equals space cos 75 degree space equals space cos left parenthesis 45 degree plus 30 degree right parenthesis space equals space cos 45 degree cos 30 degree minus sin 45 degree sin 30 degree

equals fraction numerator 1 over denominator square root of 2 end fraction. space fraction numerator square root of 3 over denominator 2 end fraction minus fraction numerator 1 over denominator square root of 2 end fraction.1 half space equals space fraction numerator 1 over denominator 2 square root of 2 end fraction left parenthesis square root of 3 minus 1 right parenthesis

Hence,

cos open parentheses fraction numerator 5 straight pi over denominator 12 end fraction close parentheses space equals space fraction numerator square root of 3 minus 1 over denominator 2 square root of 2 end fraction

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59. Find the value of:

<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>

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60.

Prove that:

$cos open parentheses fraction numerator 3 straight pi over denominator 4 end fraction plus straight x close parentheses minus space cos open parentheses fraction numerator 3 straight pi over denominator 4 end fraction minus straight x close parentheses space equals space minus square root of 2 sinx$

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