document.write( "Question 971840: what is the exact value of csc(tan-1(7x)) \n" ); document.write( "
Algebra.Com's Answer #594242 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
what is the exact value of
\n" ); document.write( "
\n" ); document.write( "
\r\n" );
document.write( "That asks the question:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "So first we draw a right triangle containing an angle θ\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Now let's make sure that the tangent of θ is 7x.  To do that\r\n" );
document.write( "we remember that \"matrix%281%2C3%2CTANGENT%2C%22%22=%22%22%2COPPOSITE%2FADJACENT%29\".\r\n" );
document.write( "Now we write 7x as the fraction \"7x%2F1\", label the OPPOSITE side\r\n" );
document.write( "of θ as the NUMERATOR of that fraction, which is 7x, and label the \r\n" );
document.write( "ADJACENT side of θ as the DENOMINATOR of that fraction, which is 1.\r\n" );
document.write( "We also find the hypotenuse of the right triangle by the Pythagorean \r\n" );
document.write( "theorem: \r\n" );
document.write( "\"c%5E2=a%5E2%2Bb%5E2\"\r\n" );
document.write( "\"c%5E2=1%5E2%2B%287x%29%5E2\"\r\n" );
document.write( "\"c%5E2=1%2B49x%5E2\"\r\n" );
document.write( "\"c=sqrt%281%2B49x%5E2%29\"\r\n" );
document.write( "\r\n" );
document.write( "and label the hypotenuse as that:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Now the problem asks:\r\n" );
document.write( "\r\n" );
document.write( "If the tangent of an angle is 7x, what is its cosecant?\r\n" );
document.write( "\r\n" );
document.write( "We remember that \"matrix%281%2C3%2CCOSECANT%2C%22%22=%22%22%2CHYPOTENUSE%2FOPPOSITE%29\".\r\n" );
document.write( "\r\n" );
document.write( "So the cosecant is \"sqrt%281%2B49x%5E2%29%2F%287x%29\".\r\n" );
document.write( "\r\n" );
document.write( "If x is positive, 7x will be positive.\r\n" );
document.write( "The tangent is positive in QI and QIII.\r\n" );
document.write( "The cosecant is positive in QI and negative in QIII.\r\n" );
document.write( "\r\n" );
document.write( "Is x is negative, 7x will be negative\r\n" );
document.write( "The tangent is negative in QII and QIV\r\n" );
document.write( "the cosecant is positive in QII and negative in QIV.\r\n" );
document.write( "\r\n" );
document.write( "Therefore to be sure to give all possible answers, and not just cases\r\n" );
document.write( "where x is positive and θ is in QI, we must put ± before\r\n" );
document.write( "the final answer:\r\n" );
document.write( "\r\n" );
document.write( "Answer:  \"%22%22+%2B-+sqrt%281%2B49x%5E2%29%2F%287x%29\".\r\n" );
document.write( "\r\n" );
document.write( "Edwin
\n" ); document.write( "
\n" );