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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.

A, B, C, D are the vertices of a cyclic quadrilateral.

$rightwards double arrow$ A + C = $180 degree space equals space straight pi$

and $straight B plus straight D space equals space 180 degree space equals space straight pi$

∴ $sin space straight A plus sin space straight B space minus space sinC space minus space sin space straight D$

 $equals space sin left parenthesis 180 degree minus straight C right parenthesis plus sin left parenthesis 180 degree minus straight D right parenthesis minus sinC minus sinD$

 $equals space sin left parenthesis straight pi minus straight C right parenthesis space plus space sin left parenthesis straight pi minus straight D right parenthesis minus sin space straight C space minus space sinD$

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(∵ x is quadrant II means sinx is +ve)

Hence, sinA + sinB - sinC - SinD = 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

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59. Find the value of:

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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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