document.write( "Question 981217: Good day, can you help me solve this problem?\r
\n" ); document.write( "\n" ); document.write( "Determine the value of β that will satisfy the ff.\r
\n" ); document.write( "\n" ); document.write( "sin (5β-10°) = 1/sec(3β+4°)\r
\n" ); document.write( "\n" ); document.write( "Thank you! \n" ); document.write( "

Algebra.Com's Answer #602369 by KMST(5328) $\"\"$ $\"About$

You can put this solution on YOUR website!
$\"sec%28A%29=1%2Fcos%28A%29\"$ <---> $\"cos%28A%29=1%2Fsec%28A%29\"$
\n" ); document.write( "Applying that,
\n" ); document.write( " $\"sin%285beta-10%5Eo%29=1%2Fsec%283beta%2B4%5Eo%29\"$ ---> $\"sin%285beta-10%5Eo%29=cos%283beta%2B4%5Eo%29\"$
\n" ); document.write( "The expressions on both sides of the equal sign must be both positive, or both negative.
\n" ); document.write( "
\n" ); document.write( "If they are both positive,
\n" ); document.write( "the angles measuring $\"5beta-10%5Eo\"$ and $\"3beta%2B4%5Eo\"$
\n" ); document.write( "are both in quadrant I.
\n" ); document.write( "
\n" ); document.write( "For two angles between $\"0%5Eo\"$ and $\"90%5Eo\"$ ,
\n" ); document.write( "if the sine of an angle and the cosine of the other are the same,
\n" ); document.write( "the angles are complementary (their measures add to $\"90%5Eo\"$ ).
\n" ); document.write( "In that case, we would have
\n" ); document.write( " $\"5beta-10%5Eo%2B3beta%2B4%5Eo=90%5Eo\"$ ---> $\"8beta-6%5Eo=90%5Eo\"$ ---> $\"8beta=96%5Eo\"$ ---> $\"beta=96%5Eo%2F6\"$ ---> $\"highlight%28beta=12%5Eo%29\"$ .
\n" ); document.write( "(That makes $\"5beta-10%5Eo=50%5Eo\"$ and $\"3beta%2B4%5Eo=40%5Eo\"$ ).
\n" ); document.write( "
\n" ); document.write( "However, since coterminal angles have the same trigonometric function values,
\n" ); document.write( "there are other solutions with both angles in the first quadrant,
\n" ); document.write( "but their measures adding to $\"90%5E0%2Bk360%5Eo\"$ , with $\"k\"$ = any integer.
\n" ); document.write( " $\"8beta-6%5Eo=90%5Eo%2Bk360%5Eo\"$ ---> $\"8beta=96%5Eo%2Bk360%5Eo\"$ ---> $\"beta=%2896%5Eo%2Bk360%5Eo%29%2F8\"$ ---> $\"beta=96%5Eo%2F8%2Bk360%5Eo%29%2F8\"$ ---> $\"highlight%28beta=12%5Eo%2Bk45%5Eo%29\"$ for any $\"k\"$ integer.
\n" ); document.write( "So, $\"57%5Eo\"$ , $\"102%5Eo\"$ , $\"147%5Eo\"$ , $\"192%5Eo\"$ , etc are solutions,
\n" ); document.write( "and so are $\"-33%5Eo\"$ , $\"-78%5Eo\"$ , $\"-123%5Eo\"$ , $\"-168%5Eo\"$ , etc.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "If $\"sin%285beta-10%5Eo%29\"$ and $\"cos%283beta%2B4%5Eo%29\"$ are both negative,
\n" ); document.write( "the angles measuring $\"5beta-10%5Eo\"$ and $\"3beta%2B4%5Eo\"$
\n" ); document.write( "are both in quadrant III.
\n" ); document.write( "They could be both be between $\"180%5E0\"$ and $\"270%5Eo\"$ ,
\n" ); document.write( "measuring $\"180%5Eo\"$ more than their quadrant I reference angles,
\n" ); document.write( "and in that case their measures would add up to
\n" ); document.write( " $\"90%5Eo%2B180%5Eo%2B180%5Eo=90%5Eo%2B360%5Eo\"$ .
\n" ); document.write( "one or both could be coterminal with angles measuring between $\"180%5E0\"$ and $\"270%5Eo\"$ ,
\n" ); document.write( "and in that case their measures would add up to
\n" ); document.write( " $\"90%5Eo%2B180%5Eo%2B180%5Eo%2Bp360%5Eo=90%5Eo%2B%28p%2B1%29360%5Eo\"$
\n" ); document.write( "In either case, the equation and solution would be the same as above:
\n" ); document.write( " $\"8beta-6%5Eo=90%5Eo%2Bk360%5Eo\"$ and $\"highlight%28beta=12%5Eo%2Bk45%5Eo%29\"$ for any $\"k\"$ integer. \n" ); document.write( "

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