document.write( "Question 1169169: Find tan, if sec=4, sin>0\r
\n" ); document.write( "\n" ); document.write( "Another question is if tan=7/4 and sin<0; what is csc and sin \n" ); document.write( "

Algebra.Com's Answer #793825 by Theo(13342) $\"\"$ $\"About$

You can put this solution on YOUR website!
problem number 1:\r
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\n" ); document.write( "\n" ); document.write( "find tan(x) if sec(x) = 4 and sine is greater than 0.\r
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\n" ); document.write( "\n" ); document.write( "tan(x) = sin(x) / cos(x)
\n" ); document.write( "sec(x) = 4
\n" ); document.write( "cos(x) = 1/sec(x) = 1/4
\n" ); document.write( "sin^2(x) + cos^2(x) = 1
\n" ); document.write( "sin^2(x) = 1 - cos^2(x) = 1 - (1/4)^2 = 1 - 1/16 = 15/16
\n" ); document.write( "sin(x) = sqrt(15/16) = sqrt(15)/4
\n" ); document.write( "tan(x) = (sqrt(15)/4)/(1/4) = (sqrt(15)/4)*4 = sqrt(15).
\n" ); document.write( "sin(x) is positive in the first and second quadrant.
\n" ); document.write( "in the first quadrant, sin(x) is positive and cos(x) is positive, therefore:
\n" ); document.write( "tan(x) = sqrt(15) in the first quadrant.
\n" ); document.write( "in the second quadrant, sin(x) is positive and cos(x) is negative, therefore:
\n" ); document.write( "tan(x) = -sqrt(15) in the second quadrant.\r
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\n" ); document.write( "\n" ); document.write( "your solution appears to be that:
\n" ); document.write( "tan(x) = sqrt(15) when x is in the first quadrant and tan(x) = -sqrt(15) when x is in the second quadrant.
\n" ); document.write( "since sin(x) is only positive in the first and second quadrant, then those quadrants are not considered.\r
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\n" ); document.write( "\n" ); document.write( "problem number 2:\r
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\n" ); document.write( "\n" ); document.write( "find sin(x) and csc(x) if tan(x) = 7/4) and sine is less than 0.\r
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\n" ); document.write( "\n" ); document.write( "tan(x) = (7/4)
\n" ); document.write( "tangent function = opposite side / adjacent side.
\n" ); document.write( "when tan(x) = 7/4, opposite side = 7 and adjacent side = 4.
\n" ); document.write( "hypotenuse = sqrt(7^2 + 4^2) = sqrt(65)
\n" ); document.write( "sin(x) = opposite side / hypotenuse = 7/sqrt(65)
\n" ); document.write( "tan(x) is positive in the first and third quadrants only.
\n" ); document.write( "in the first quadrant, sin(x) is positive, so first quadrant is not good.
\n" ); document.write( "in the third quadrant, sin(x) is negative, so third quadrant is good.
\n" ); document.write( "therefore, tan(x) = 7/4 in the third quadrant only if sin(x) has to be less than 0.
\n" ); document.write( "in the third quadrant, when tan(x) = 7/4, .....
\n" ); document.write( "sin(x) = -7/sqrt(65)
\n" ); document.write( "csc(x) = 1/sin(x) = 1/(-7/sqrt(65) = -sqrt(65)/7.\r
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\n" ); document.write( "\n" ); document.write( "your solution appears to be that:\r
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\n" ); document.write( "\n" ); document.write( "when tan(x) = 7/4) and sine is less than 0, then sin(x) = -7/sqrt(65) and csc(x) = -sqrt(65)/7\r
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\n" ); document.write( "\n" ); document.write( "let me know if you have any questions or concerns about these answers.\r
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\n" ); document.write( "\n" ); document.write( "theo\r
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