document.write( "Question 934973: Simplify without using tables or calculators.sin^2 495degrees(1-tan^2 780degrees/(1 cos^2 600degrees)(1 tan^ 570degrees) \n" ); document.write( "

Algebra.Com's Answer #568025 by KMST(5328) $\"\"$ $\"About$

You can put this solution on YOUR website!
I suspect that what you mean may have been
\n" ); document.write( "sin^2 (495degrees)(1-tan^2 (780degrees))/(1 - cos^2 (600degrees))(1-tan^2) 570degrees))=

,
\n" ); document.write( "but whatever the operations among those trigonometric functions,
\n" ); document.write( "the expression's value is much simpler than it looks.
\n" ); document.write( "The key is to find a more familiar/convenient co-terminal angle (one that differs by one or more turns in either direction). I aim for one between $\"-180%5Eo\"$ and $\"-180%5Eo\"$ .
\n" ); document.write( "From there, you look for a reference angle in the first quadrant.
\n" ); document.write( "The reference angle is one with a symmetrical terminal side. All trigonometric have the same absolute value for the reference angle, so you just have to worry about the sign. (And if all ther trigonometric functions in your expression are squared, you do not even have to worry about the sign.
\n" ); document.write( "
\n" ); document.write( " $\"495%5Eo=360%5Eo%2B135%5Eo\"$ so $\"sin%28495%5Eo%29=sin%28135%5Eo%29\"$ ,
\n" ); document.write( "and $\"135%5Eo\"$ is an angle in the second quadrant
\n" ); document.write( "that is symmetrical to $\"180%5Eo-135%5Eo=45%5Eo\"$ ,
\n" ); document.write( "so $\"sin%28495%5Eo%29=sin%28135%5Eo%29=sin%2845%5Eo%29=sqrt%282%29%2F2=sqrt%281%2F2%29\"$ ,
\n" ); document.write( "and $\"sin%5E2+%28495%5Eo%29=%28sqrt%281%2F2%29%29%5E2=1%2F2\"$
\n" ); document.write( "
\n" ); document.write( " $\"780%5Eo=720%5Eo%2B60%5Eo=2%28360%5Eo%29%2B60%5Eo\"$ so $\"tan%28780%5Eo%29=tan%2860%5Eo%29=sqrt%283%29\"$ ,
\n" ); document.write( "so $\"tan%5E2+%28780%5Eo%29=%28sqrt%283%29%29%5E2=3\"$ ,
\n" ); document.write( "and $\"%281-tan%5E2+%28780%5Eo%29%29=1-3=-2\"$ .
\n" ); document.write( "
\n" ); document.write( " $\"600%5Eo=720%5Eo-120%5Eo\"$ so $\"cos%28600%5Eo%29=cos%28-120%5Eo%29\"$ ,
\n" ); document.write( "and $\"-120%5Eo\"$ is an angle in the third quadrant,
\n" ); document.write( "that is symmetrical to $\"120%5Eo\"$ in the second quadrant,
\n" ); document.write( "and to $\"180%5Eo-120%5Eo=60%5Eo\"$ in the first quadrant.
\n" ); document.write( " $\"cos%28600%5Eo%29=cos%28-120%5Eo%29=cos%28120%5Eo%29=-cos%2860%5Eo%29=-1%2F2\"$ .
\n" ); document.write( "Then, $\"cos%5E2%28600%5Eo%29=%28-1%2F2%29%5E2=1%2F4\"$ and $\"1-cos%5E2%28600%5Eo%29=1-1%2F4=3%2F4\"$ .
\n" ); document.write( "
\n" ); document.write( " $\"570%5Eo=360%5Eo%2B210%5Eo\"$ and $\"210%5Eo=180%5Eo%2B30%5Eo\"$ ,
\n" ); document.write( "so the reference angle for $\"570%5Eo+\"$ is $\"30%5Eo\"$ ,
\n" ); document.write( "and $\"tan%28570%5Eo%29=tan%2830%5Eo%29=sqrt%283%29%2F3=sqrt%281%2F3%29\"$ .
\n" ); document.write( "Then, $\"tan%5E2%28570%5Eo%29=%28sqrt%281%2F3%29%29%5E2=1%2F3\"$ and
\n" ); document.write( " $\"1-tan%5E2%28570%5Eo%29=1-1%2F3=2%2F3\"$ .
\n" ); document.write( "
\n" ); document.write( "So if your expression was

,
\n" ); document.write( "

\n" ); document.write( "

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