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Trigonometric Functions

Short Answer Type

101.

Find the value of:

$cos straight pi over 8$

109 Views

Long Answer Type

102.

Prove that:

$tan space straight A space plus space tan space left parenthesis 60 degree plus straight A right parenthesis minus tan space left parenthesis 60 degree minus straight A right parenthesis space equals space 3 space tan space 3 straight A.$

134 Views

Short Answer Type

103.
Show that:

$cos straight pi over 7. space cos fraction numerator 2 straight pi over denominator 7 end fraction. space cos fraction numerator 4 straight pi over denominator 7 end fraction space equals space minus 1 over 8$

L.H.S. = $cos straight pi over 7. cos fraction numerator 2 straight pi over denominator 7 end fraction. cos fraction numerator 4 straight pi over denominator 7 end fraction space equals space fraction numerator 1 over denominator 2 sin begin display style straight pi over 7 end style end fraction open parentheses 2 sin straight pi over 7 cos straight pi over 7 close parentheses. cos fraction numerator 2 straight pi over denominator 7 end fraction cos fraction numerator 4 straight pi over denominator 7 end fraction$

$equals space fraction numerator 1 over denominator 2 sin open parentheses begin display style straight pi over 7 end style close parentheses end fraction cross times sin open parentheses fraction numerator 2 straight pi over denominator 7 end fraction close parentheses space cos open parentheses fraction numerator 2 straight pi over denominator 7 end fraction close parentheses space cos open parentheses fraction numerator 4 straight pi over denominator 7 end fraction close parentheses$ (∵ $2 space sinθ space space cosθ space equals space sin space 2 straight theta$ )

$equals fraction numerator 1 over denominator 4 space sin open parentheses begin display style straight pi over 7 end style close parentheses end fraction cross times open square brackets 2 space sin open parentheses fraction numerator 2 straight pi over denominator 7 end fraction close parentheses space cos fraction numerator 2 straight pi over denominator 7 end fraction close square brackets space cos fraction numerator 4 straight pi over denominator 7 end fraction$

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$equals fraction numerator 1 over denominator 8 sin open parentheses begin display style straight pi over 7 end style close parentheses end fraction cross times sin open parentheses straight pi plus straight pi over 7 close parentheses space equals space fraction numerator 1 over denominator 8 sin open parentheses begin display style straight pi over 7 end style close parentheses end fraction cross times space minus sin open parentheses straight pi over 7 close parentheses$ [∵ $sin left parenthesis straight pi plus straight theta right parenthesis space equals space minus sinθ$ ]

$space space equals 1 over 8 cross times negative 1 space equals space minus 1 over 8 space equals space straight R. straight H. straight S.$

Hence, L.H.S. = R.H.S.

131 Views

Long Answer Type

104.

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

$tanx space equals space minus 4 over 3 comma space straight x space is space in space quadrant space II$

135 Views

105.

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

$cosx space equals space minus 1 third comma space straight x$ is in quadrant III

135 Views

Short Answer Type

106.

If $cosx space equals space 1 half open parentheses straight a plus 1 over straight a close parentheses comma$ show that: $cos 2 straight x space equals space 1 half open parentheses straight a squared plus 1 over straight a squared close parentheses$

108 Views

107.

If cos x = $1 half open parentheses straight a plus 1 over straight a close parentheses comma$ show that:

$cos 3 straight x space equals space 1 half open parentheses straight a cubed plus 1 over straight a cubed close parentheses$

127 Views

108.

Prove the following:

$fraction numerator sin 5 straight x plus sin 3 straight x over denominator cos 5 straight x plus cos 3 straight x end fraction space equals space tan 4 straight x$

123 Views

109.

Prove the following:

$fraction numerator sin space straight x minus sin space straight y over denominator cos space straight x plus cos space straight y end fraction space equals space tan open parentheses fraction numerator straight x minus straight y over denominator 2 end fraction close parentheses$

130 Views

110.

Prove the following:

$sin open parentheses fraction numerator 7 straight pi over denominator 12 end fraction close parentheses plus cos open parentheses fraction numerator 7 straight pi over denominator 12 end fraction close parentheses space equals space cos open parentheses straight pi over 4 close parentheses space equals fraction numerator 1 over denominator square root of 2 end fraction$