document.write( "Question 808521: Find all values of x in the interval [0, 2π] that satisfy the given equation.\r
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\n" ); document.write( "\n" ); document.write( "(sin x)^2 - 12 cos x - 12 = 0 \n" ); document.write( "
Algebra.Com's Answer #487057 by DrBeeee(684)\"\" \"About 
You can put this solution on YOUR website!
Given:
\n" ); document.write( "(1) \"sin%5E2%28x%29+-+12cosx+-12+=+0\"
\n" ); document.write( "Use the trig identity
\n" ); document.write( "(2) \"sin%5E2%28x%29+%2B+cos%5E2%28x%29+=+1\" to get
\n" ); document.write( "(3) \"sin%5E2%28x%29+=+1+-+cos%5E2%28x%29\"
\n" ); document.write( "Put (3) into (1) to obtain
\n" ); document.write( "(4) \"1+-+cos%5E2%28x%29+-+12cosx+-12+=+0\" or
\n" ); document.write( "(5) \"-+cos%5E2%28x%29+-+12cosx+-+11+=+0\" or
\n" ); document.write( "(6) \"cos%5E2%28x%29+%2B+12cosx+%2B+11+=+0\"
\n" ); document.write( "which factors into
\n" ); document.write( "(7)\"%28cos%28x%29+%2B+1%29%2A%28cosx+%2B+11%29+=+0\"
\n" ); document.write( "Now set each factor equal to zero and solve for x, and get
\n" ); document.write( "(8)\"cos%28x%29+%2B+11+=+0\" and
\n" ); document.write( "(9)\"cosx+%2B+1+=+0\" or
\n" ); document.write( "(10)\"cosx+=+-11\" and
\n" ); document.write( "(11)\"cosx+=+-1\"
\n" ); document.write( "Since absolute value of cos(x) is never greater than one, (10) has no solutions, whereas (12) is
\n" ); document.write( "(13) x = arccos(-1) or
\n" ); document.write( "(14) x = 180 degrees
\n" ); document.write( "Let's check this with (1).
\n" ); document.write( "Is (sin(180)^2 -12cos(180) -12 = 0)?
\n" ); document.write( "Is (0^2 - 12*(-1) - 12 = 0)?
\n" ); document.write( "Is (0 + 12 - 12 = 0)?
\n" ); document.write( "Is (0 = 0)? Yes
\n" ); document.write( "Answer: the only x in the interval [0,360] is x = 180 degrees or \"pi\" radians.
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