document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1004654>Question 1004654</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 1.   Use table of values and plotting points in polar grid to graph the polar equation r=2/sin(theta).\r<BR>\n" );
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document.write( "2.   What type of graph did you get?\r<BR>\n" );
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document.write( "3.   Justify your answer to part 2 by converting polar equation to rectangular equation.\r<BR>\n" );
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document.write( "I HAVE FILLED IN THE VALUES BELOW.\r<BR>\n" );
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document.write( "0             r=2/sin(theta)<BR>\n" );
document.write( "pi/6          undefined<BR>\n" );
document.write( "pi/4          4<BR>\n" );
document.write( "pi/3          2sqroott2<BR>\n" );
document.write( "pi/2          4sqroot3/3<BR>\n" );
document.write( "2pi/3         2<BR>\n" );
document.write( "3pi/4         4sqroot3/3<BR>\n" );
document.write( "5pi/6         4<BR>\n" );
document.write( "pi            undefined<BR>\n" );
document.write( "3pi/2         -2<BR>\n" );
document.write( "2pi           undefined\r<BR>\n" );
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document.write( "When I start looking at it and trying to graph I get so confused. I am seeing everything from a circle on a graphing calculator, to a straight line, and even a parabola.  Can anyone help me with the answers and graph? Maybe point out where I went wrong? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #621124 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=621124\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the equation in polar coordinates is r = 2/sin(theta)\r<BR>\n" );
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document.write( "in rectangular coordinates, you would get:\r<BR>\n" );
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document.write( "x = r * cos(theta)<BR>\n" );
document.write( "y = r * sin(theta)\r<BR>\n" );
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document.write( "using pythagorus, the hypotenuse of the triangle formed by x and y is equal to:\r<BR>\n" );
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document.write( "h^2 = x^2 + y^2 = r^2 * cos^2(theta) + r^2 * sin^2(theta) which becomes:\r<BR>\n" );
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document.write( "h^2 = r^2 * (sin^2(theta) + cos^2(theta)) which becomes:\r<BR>\n" );
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document.write( "h^2 = r^2 which becomes:\r<BR>\n" );
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document.write( "hypotenuse = j = r\r<BR>\n" );
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document.write( "you are given that r = 2 / sin(theta)\r<BR>\n" );
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document.write( "this means that the hypotenuse is equal to 2 / sin(theta).\r<BR>\n" );
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document.write( "the sine of theta is equal to y / r\r<BR>\n" );
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document.write( "r is equal to 2 / sin(theta)\r<BR>\n" );
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document.write( "this means that r is equal to 2 / (y / r)\r<BR>\n" );
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document.write( "since 2 / (y / r) is the same as 2 * (r / y), then:\r<BR>\n" );
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document.write( "r is equal to 2 * (r / y)\r<BR>\n" );
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document.write( "multiply both sides of this equation by y and divide both sides of this equation by r and you get:\r<BR>\n" );
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document.write( "y = 2\r<BR>\n" );
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document.write( "what this says is that y is going to be equal to 2 regardless of the value of x.\r<BR>\n" );
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document.write( "that means that the graph of the equation of r = 2 / sin(theta) is a straight line that is parallel to the x-axis and has a value of 2.\r<BR>\n" );
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document.write( "you can verify this graphically.\r<BR>\n" );
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document.write( "just plot the graph of y = 2 / sin(theta) and you will see that it is a straight line at y = 2.\r<BR>\n" );
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document.write( "you need graphing software that can graph equations in polar form.\r<BR>\n" );
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document.write( "the following calculator can do that:\r<BR>\n" );
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document.write( "<a  rel=nofollow HREF= \"http://www.desmos.com/calculator\" target = \"_blank\">http://www.desmos.com/calculator</a>\r<BR>\n" );
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document.write( "the graph i made is shown below:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2015/111805.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( "you can see that the graph is a horizontal line at y = 2.\r<BR>\n" );
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document.write( "the following picture may help clarify the relationship between the polar coordinate form and the rectangular form of the equation.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2015/111806.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( "what you see in the pictures is that r = 2 / sin(theta) is the hypotenuse of a right triangle that has theta as the angle and the opposite side as y and the adjacent side as x.\r<BR>\n" );
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document.write( "since sin(theta) = y/r, then y = r * sin(theta)\r<BR>\n" );
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document.write( "since cos(theta) = x/r, then x = r * cos(theta)\r<BR>\n" );
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document.write( "since r is given as being equal to 2/sin(theta), you can replace r in the equations for x and y with 2/sin(theta) and solve for x and y.\r<BR>\n" );
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document.write( "you get y = 2\r<BR>\n" );
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document.write( "you get x = 2 cot(theta).\r<BR>\n" );
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document.write( "by the pythagorean formula, r = sqrt(x^2 + y^2)\r<BR>\n" );
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document.write( "replace x^2 with (2cot(theta)^2 and replace y with 2^2 and you wind up with:\r<BR>\n" );
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document.write( "r = sqrt(4cot^2(theta) + 4)\r<BR>\n" );
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document.write( "this is also equivalent to r = sqrt(4/tan^2(theta) + 4)\r<BR>\n" );
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document.write( "that last form is easier to calculate from when you use your calculator.\r<BR>\n" );
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document.write( "what i think you were trying to do is manually plot the equation.\r<BR>\n" );
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document.write( "that's a chore.\r<BR>\n" );
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document.write( "you can do it, but it's much easier to graph it using appropriate graphing software.\r<BR>\n" );
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document.write( "also knowing the translations from polar to rectangular helps.\r<BR>\n" );
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document.write( "i used graphing software to see that the graph was a line at y = 2.\r<BR>\n" );
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document.write( "once i saw that, i was able to figure out how it became that way using algebra.\r<BR>\n" );
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document.write( "calculating for y said that y was always going to be equal to 2.\r<BR>\n" );
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document.write( "that corresponded with the graph and so i was reasonably confident that the equation of r = 2 / sin(theta) was the same as the equation y = 2.\r<BR>\n" );
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document.write( "what you were doing was trying to calculate r which would not be the same each time but would always wind up creating a line where the y value of the coordinate point would be equal to 2.\r<BR>\n" );
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document.write( "you would only have been able to see this after connecting all those points.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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