document.write( "Question 1125743: FInd the values of θ between 0° and 180° for which
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document.write( "a) secθ= cosecθ
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document.write( "b) tanθ= -1
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document.write( "c) sinθ - sqrt(3) cosθ = 0 \n" );
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Algebra.Com's Answer #742050 by Theo(13342)![]() ![]() You can put this solution on YOUR website! it helps to put all this stuff in the first quadrant and then figure out what quadrant it really belongs in. \n" ); document.write( "so far, that's worked ok for me.\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "i'll use x instead of theta. \n" ); document.write( "anwswer will be the same even though the variable name used is different. \n" ); document.write( "theta means the same as x. \n" ); document.write( "using x makes graphing a little easier.\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "problem A.\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "start with sec(x) = csc(x) \n" ); document.write( "sec(x) is equal to 1/cos(x) \n" ); document.write( "csc(x) is equal to 1/sin(x) \n" ); document.write( "therefore 1/sin(x) = 1/cos(x) \n" ); document.write( "cross multiply to get xin(x) = cos(x) \n" ); document.write( "divide both sides of this equation by cos(x) to get: \n" ); document.write( "sin(x)/cos(x) = 1 \n" ); document.write( "sin(x)/cos(x) = tan(x) \n" ); document.write( "therefore tan(x) = 1 \n" ); document.write( "that makes x = arctan(1) = 45 degrees. \n" ); document.write( "that's in the first quadrant. \n" ); document.write( "tangent is positive in the first quadrant and the third quadrant. \n" ); document.write( "0 to 180 degrees is in the first quadrant and the second quadrant. \n" ); document.write( "therefore x = 45 degrees in the interval between 0 and 180 degrees. \n" ); document.write( "here's the graph of y = sec(x) and the graph fo y = csc(x). \n" ); document.write( "their intersection is equal to 45 degrees in the interval between x = 0 and x = 180 degrees, confirming the solution is correct. \n" ); document.write( "the following graph confirms the solution is correct. \n" ); document.write( " ![]() \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "problem B.\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "start with tan(x) = -1 \n" ); document.write( "place it in the first quadrant by making it positive. \n" ); document.write( "you get tan(x) = 1. \n" ); document.write( "this makes x = arctan(1) which makes x = 45 degrees. \n" ); document.write( "your angle is 45 degrees in the first quadrant. \n" ); document.write( "but you want tan(x) to be equal to -1. \n" ); document.write( "tan(x) is positive in the first and third quadrant, but negative in the second and fourth quadrant. \n" ); document.write( "therefore, your angle has to be in the second quadrant. \n" ); document.write( "the equivalent angle in the second quadrant is equal to 180 - 1 = 135 degrees. \n" ); document.write( "the angle that gives you tan(x) = -1 in the interval between 0 and 180 degrees is therefore 135 degrees. \n" ); document.write( "the following graph confirms the solution is correct. \n" ); document.write( " ![]() \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "problem C.\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "start with sin(x) - sqrt(3) * cos(x) = 0 \n" ); document.write( "add sqrt(3) * cos(x) to both sides of the equation to get: \n" ); document.write( "sin(x) = sqrt(3) * cos(x) \n" ); document.write( "divide both sides of this equation by cos(x) toget: \n" ); document.write( "sin(x) / cos(x) = sqrt(3) \n" ); document.write( "sin(x) / cos(x) = tan(x) \n" ); document.write( "equation becomes tan(x) = sqrt(3). \n" ); document.write( "this makes x = arctan(srt(3) which makes x = 60 degrees. \n" ); document.write( "tan(x) is positive in the first and third quadrants, therefore: \n" ); document.write( "the solution is x = 60 degrees in the interval between 0 and 180 degrees. \n" ); document.write( "the following graph confirms the solution is correct. \n" ); document.write( " ![]() \n" ); document.write( " \n" ); document.write( " \n" ); document.write( " \n" ); document.write( " \n" ); document.write( " \n" ); document.write( " \n" ); document.write( "\n" ); document.write( " \n" ); document.write( " |