document.write( "Question 840926: Find the value of sec(theta) if tan(theta)= Negative square root of 114 divided by 19 and sin(theta)>0. \n" ); document.write( "
Algebra.Com's Answer #506503 by Edwin McCravy(20060)\"\" \"About 
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Find the value of sec(θ) if tan(θ)= \"-sqrt%28114%29%2F19\" and sin(θ)>0.
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document.write( "There are two ways to do this problem.  We'll do it both ways:\r\n" );
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document.write( "First way.  θ is in quadrant II because that \r\n" );
document.write( "is the only quadrant in which the tangent \r\n" );
document.write( "is a negative number and the sine is a \r\n" );
document.write( "positive number.\r\n" );
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document.write( "So we draw the angle θ in quadrant II.\r\n" );
document.write( "The tangent = \"%28opposite%29%2F%28adjacent%29\" = \"y%2Fx\". So\r\n" );
document.write( "we take the numerator of the tangent \"%22%22%2Bsqrt%28114%29\" as y, \r\n" );
document.write( "(we take it positive because it goes up from the x-axis).\r\n" );
document.write( "And we take the denominator of the tangent -19 as x, \r\n" );
document.write( "(we take it negative because it goes left from the origin).\r\n" );
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document.write( "We use the Pythagorean theorem: \"hypotenuse%5E2=adjacent%5E2%2Bopposite%5E2\"\r\n" );
document.write( "or rē = xē + yē\r\n" );
document.write( "   rē = 19ē + \"%28sqrt%28114%29%29%5E2\"\r\n" );
document.write( "   rē = 361 + 114\r\n" );
document.write( "   rē = 475\r\n" );
document.write( "    r = \"sqrt%28475%29\" = \"sqrt%2825%2A19%29\" = \"5sqrt%2819%29\"\r\n" );
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document.write( "(r is always taken positive).\r\n" );
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document.write( "And since we know that the secant = \"hypotenuse%2Fadjacent\" = \"r%2Fx\",\r\n" );
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document.write( "sec(θ) = \"%285sqrt%2819%29%29%2F%28-19%29\" = \"-5sqrt%2819%29%2F19\"\r\n" );
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document.write( "Second way:\r\n" );
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document.write( "Use the identity: \r\n" );
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document.write( "1 + tanē(θ) = secē(θ)\r\n" );
document.write( "1 + \"%28-sqrt%28114%29%2F19%29%5E2\" = secē(θ)\r\n" );
document.write( "1 + \"114%2F361\" = secē(θ)\r\n" );
document.write( "\"361%2F361\" + \"114%2F361\" = secē(θ)\r\n" );
document.write( "\"475%2F361\" = secē(θ)\r\n" );
document.write( "\"%22%22+%2B-+sqrt%28475%2F361%29\" = sec(θ)\r\n" );
document.write( "\"%22%22+%2B-+sqrt%28475%29%2Fsqrt%28361%29\" = sec(θ) \r\n" );
document.write( "\"%22%22+%2B-+sqrt%2825%2A19%29%2F19\" = sec(θ)\r\n" );
document.write( "\"%22%22+%2B-+5sqrt%2819%29%2F19\" = sec(θ)\r\n" );
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document.write( "We have to decide which quadrant θ is in just\r\n" );
document.write( "as we did using the first method.  So the secant\r\n" );
document.write( "in QII is negative, so the answer is\r\n" );
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document.write( "\"-+5sqrt%2819%29%2F19\" = sec(θ)\r\n" );
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document.write( "You can do the problem either way.\r\n" );
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document.write( "Edwin
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