document.write( "Question 683115: Please solve equation for x over the interval [0°,360°). {{3sectheta-2√3=0}}
\n" ); document.write( "show all work. \n" ); document.write( "

Algebra.Com's Answer #423485 by jsmallt9(3758) $\"\"$ $\"About$

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$\"3sec%28theta%29-2sqrt%283%29=0\"$
\n" ); document.write( "First, isolate the function. Adding $\"2sqrt%283%29\"$ :
\n" ); document.write( " $\"3sec%28theta%29=2sqrt%283%29\"$
\n" ); document.write( "And divide by 3:
\n" ); document.write( " $\"sec%28theta%29=2sqrt%283%29%2F3\"$
\n" ); document.write( "Perhaps if you know your special angle values very well you might recognize that the right side is a special angle value for sec. If not, then we can convert this to cos. Since cos and sec are reciprocals of each other then if
\n" ); document.write( " $\"sec%28theta%29=2sqrt%283%29%2F3\"$
\n" ); document.write( "then
\n" ); document.write( " $\"cos%28theta%29=3%2F%282sqrt%283%29%29\"$
\n" ); document.write( "Rationalizing the right side:
\n" ); document.write( " $\"cos%28theta%29=%283%2F%282sqrt%283%29%29%29%28sqrt%283%29%2Fsqrt%283%29%29\"$
\n" ); document.write( " $\"cos%28theta%29=%283sqrt%283%29%29%2F%282%2A3%29\"$
\n" ); document.write( " $\"cos%28theta%29=sqrt%283%29%2F2\"$
\n" ); document.write( "This is obviously a special angle value for cos. The references angle whose cos is $\"sqrt%283%29%2F2\"$ is 30. Since cos (and sec) are positive in the 1st and 4th quadrants we get a general solution of:
\n" ); document.write( " $\"theta+=+30+%2B+360n\"$
\n" ); document.write( " $\"theta+=+-30+%2B+350n\"$
\n" ); document.write( "This general solution expresses the infinite number of values for $\"theta\"$ that are solutions to your original equation.

\n" ); document.write( "Your problem asks for solutions over the interval [0, 360). For these solutions we take the general solution equations and replace the n's with integers until we are satisfied we have found all the $\"theta\"$ 's in the interval.
\n" ); document.write( "From the equation:
\n" ); document.write( " $\"theta+=+30+%2B+360n\"$
\n" ); document.write( "if n = 0 then $\"theta+=+30\"$
\n" ); document.write( "if n = 1 (or higher integers), $\"theta\"$ is more than 360.
\n" ); document.write( "if n = -1 (or any other negative integer), $\"theta\"$ is below 0.
\n" ); document.write( "From the equation:
\n" ); document.write( " $\"theta+=+-30+%2B+360n\"$
\n" ); document.write( "if n = 0 (or any negative integer), $\"theta\"$ is below 0.
\n" ); document.write( "if n = 1, $\"theta\"$ is 330.
\n" ); document.write( "if n = 2 (or higher integers), $\"theta\"$ is more than 360.
\n" ); document.write( "So the only solutions over the interval [0, 360) are 30 and 330. \n" ); document.write( "

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