document.write( "Question 429870: cos(x+(π/2))-cos(2x)=0 \n" ); document.write( "

Algebra.Com's Answer #298549 by Gogonati(855) $\"\"$ $\"About$

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Solution:From the trigonometry we know that: $\"cos2x=%28cos%28x%29%29%5E2-%28sin%28x%29%29%5E2\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"%28cos%28x%29%29%5E2=1-%28sin%28x%29%29%5E2\"$ , and cos(90+x)=-sin(x). Now we write:\r
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\n" ); document.write( "\n" ); document.write( " $\"2%28sin%28x%29%29%5E2-sin%28x%29-1=0\"$ , substitute six=a and rewrite the equation:\r
\n" ); document.write( "\n" ); document.write( " $\"2a%5E2-a-1=0\"$ , which is a quadratic equation. Solve this equation:\r
\n" ); document.write( "\n" ); document.write( "There are two roots: a=1 and a=-1/2, thus we have: sin(x)=1 and sin(x)=-12\r
\n" ); document.write( "\n" ); document.write( "Solve now these two trigonometric equations and find:\r
\n" ); document.write( "\n" ); document.write( "x=2n* $\"pi\"$ + $\"pi\"$ /2, \r
\n" ); document.write( "\n" ); document.write( "x=2n* $\"pi\"$ - $\"pi\"$ /6, and x=2n* $\"pi\"$ + $\"7pi\"$ /6\r
\n" ); document.write( "\n" ); document.write( "Done.\r
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