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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:

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

L.H.S. = $tan space straight A space plus space tan left parenthesis 60 degree plus straight A right parenthesis minus tan left parenthesis 60 degree minus straight A right parenthesis$

= $tanA space plus space fraction numerator tan space 60 degree plus tan space straight A over denominator 1 minus tan space 60 degree tan space straight A end fraction minus fraction numerator tan space 60 degree minus tan space straight A over denominator 1 plus tan space 60 degree tan space straight A end fraction$

= $tan space straight A plus fraction numerator square root of 3 plus tan space straight A over denominator 1 minus square root of 3 tan space straight A end fraction minus fraction numerator square root of 3 minus tan space straight A over denominator 1 plus square root of 3 tan space straight A end fraction$

= $tanA space plus space fraction numerator left parenthesis square root of 3 plus tan space straight A right parenthesis space left parenthesis 1 plus square root of 3 space tanA right parenthesis space minus space left parenthesis square root of 3 minus tanA right parenthesis left parenthesis 1 minus square root of 3 tanA right parenthesis over denominator left parenthesis 1 minus square root of 3 tanA right parenthesis space left parenthesis 1 plus square root of 3 tanA right parenthesis end fraction$

= $space space space tan space straight A space plus space fraction numerator square root of 3 plus 3 space tanA plus tanA plus square root of 3 tan squared straight A space minus space square root of 3 space plus space 3 tanA plus tanA minus square root of 3 tan squared straight A over denominator 1 minus 3 tan squared straight A end fraction$

$equals fraction numerator tan space straight A left parenthesis 1 minus 3 tan squared straight A right parenthesis plus 8 space tan space straight A over denominator 1 minus 3 tan squared straight A end fraction space equals space fraction numerator tan space straight A minus 3 tan cubed space straight A plus 8 space tanA over denominator 1 minus 3 space tan squared straight A end fraction$

$space space equals fraction numerator 9 space tan space straight A minus 3 tan cubed straight A over denominator 1 minus 3 tan squared space straight A end fraction space equals space 3 open parentheses fraction numerator 3 space tan space straight A minus tan cubed straight A over denominator 1 minus 3 tan squared space straight A end fraction close parentheses$

= 3 tan 3A $open square brackets because space space tan 3 straight A space equals space fraction numerator 3 tanA minus tan cubed straight A over denominator 1 minus 3 tan squared straight A end fraction close square brackets$

= R.H.S.

∴ L.H.S. = R.H.S.

Hence, $tanA space plus space tan left parenthesis 60 degree plus straight A right parenthesis space minus space tan left parenthesis 60 degree minus straight A right parenthesis space equals space 3 space tan space 3 straight A$

134 Views

Short Answer Type

103.

Show that:

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

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:

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