document.write( "Question 747138: What are the x's of: ((sin^2)x) +(tan)x) -1)) = 0 \n" ); document.write( "

Algebra.Com's Answer #454739 by KMST(5397) $\"\"$ $\"About$

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$\"%28sin%28x%29%29%5E2%2Btan%28x%29-1=0\"$ <--> $\"1-%28cos%28x%29%29%5E2%2Btan%28x%29-1=0\"$ <--> $\"-%28cos%28x%29%29%5E2%2Btan%28x%29=0\"$ <--> $\"%28cos%28x%29%29%5E2=tan%28x%29\"$ <--> $\"%28cos%28x%29%29%5E2=sin%28x%29%2Fcos%28x%29\"$ <--> $\"%28cos%28x%29%29%5E3=sin%28x%29\"$
\n" ); document.write( " $\"%28cos%28x%29%29%5E3=sin%28x%29\"$ --> $\"%28cos%28x%29%29%5E6=%28sin%28x%29%29%5E2\"$ --> $\"%28cos%28x%29%29%5E6=1-%28cos%28x%29%29%5E2\"$
\n" ); document.write( "Calling $\"%28cos%28x%29%29%5E2=y\"$ we can re-write that equation as
\n" ); document.write( " $\"y%5E3=1-y\"$ <--> $\"y%5E3%2By-1=0\"$
\n" ); document.write( "The only real solution is approximately $\"y=0.6823278038\"$
\n" ); document.write( "Going back to $\"x\"$ we have $\"%28cos%28x%29%29%5E2=0.6823278038\"$
\n" ); document.write( "Between $\"0%5Eo\"$ and $\"360%5Eo\"$ (between $\"0\"$ and $\"2pi\"$ radians) there are 4 angles with $\"%28cos%28x%29%29%5E2=0.6823278038\"$ .
\n" ); document.write( "However, solutions to $\"%28cos%28x%29%29%5E2=tan%28x%29\"$ require positive tangent, so the only solutions will be in the first and third quadrants.
\n" ); document.write( " $\"x=34.3%5Eo\"$ (or $\"x=0.598767\"$ radians) is the approximate solution in the first quadrant.
\n" ); document.write( " $\"x=34.3%5Eo%2B180%5Eo=214.3%5Eo\"$ (or $\"x=0.598767%2Bpi=3.740539\"$ radians) is the approximate solution in the third quadrant. \n" ); document.write( "

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