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Trigonometric Functions

Short Answer Type

581.

Find the general expression for all angles whose sine is zero.

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582. Find the general expression for all angles whose tangent is zero.

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583.

Find the general solution or the general expression for all angles, whose cosine is zero.

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584.

Find the general expression for all the angles whose cotangent is zero.

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585. Find a general expression for all angles having the same sine, i.e. when sin x = sin y, find the general value of x

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586.
Find general value of x, when cos x = cos y

We have cosx = cosy or cosx - cosy = 0

$rightwards double arrow$ $negative 2 sin open parentheses fraction numerator straight x plus straight y over denominator 2 end fraction close parentheses sin open parentheses fraction numerator straight x minus straight y over denominator 2 end fraction close parentheses equals 0$

$rightwards double arrow$ either $space sin fraction numerator straight x plus straight y over denominator 2 end fraction equals 0$ or $sin fraction numerator straight x minus straight y over denominator 2 end fraction equals 0$

$rightwards double arrow$ either $fraction numerator straight x plus straight y over denominator 2 end fraction space equals space kπ$ or $fraction numerator straight x minus straight y over denominator 2 end fraction space equals space kπ comma$ where $straight k space element of space straight Z$

$rightwards double arrow$ either $space space straight x plus straight y equals 2 kπ$ or $straight x minus straight y equals 2 kπ comma$ where $straight k space element of space straight Z$

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

$rightwards double arrow$ $straight x equals 2 kπ plus-or-minus straight y comma$ where $straight k space element of space straight Z$

Hence, the general solution is $straight x equals 2 nπ plus-or-minus straight y comma$ $straight n element of space straight Z$

COROLLARY: If secx = secy $rightwards double arrow$ cosx = cosy

So, when secx = secy, the general solution of $straight x equals 2 nπ plus-or-minus straight y comma$ where $straight n element of space straight Z$