document.write( "Question 1044780: Please give the corresponding equations in rectangular coordinates? \r
\n" ); document.write( "\n" ); document.write( "1) r = π
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\n" ); document.write( "3) r = 4/(3cosθ -sinθ )
\n" ); document.write( "4) r^2 = 36/(9-13sin^2(θ) )\r
\n" ); document.write( "\n" ); document.write( "I would be very grateful for a short explanation for each answer. Thank you. \n" ); document.write( "
Algebra.Com's Answer #660164 by Edwin McCravy(20065) $\"\"$ $\"About$
You can put this solution on YOUR website!
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document.write( "1) r = π\r\n" );
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document.write( "Notice that that is the same as r = 3.1416...\r\n" );
document.write( "Even though π is usually a value of θ, not r, it\r\n" );
document.write( "is a value of r in this case. Think of the equation as\r\n" );
document.write( "if it were \r\n" );
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document.write( "   r = 3.14... + (0)θ\r\n" );
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document.write( "To plot the graph in polar coordinates we make a \r\n" );
document.write( "table of values:\r\n" );
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document.write( "  θ | r\r\n" );
document.write( "  0 | 3.14...\r\n" );
document.write( " π/3| 3.14...\r\n" );
document.write( " π/2| 3.14...\r\n" );
document.write( "2π/3| 3.14...\r\n" );
document.write( "   π| 3.14...\r\n" );
document.write( "4π/3| 3.14...\r\n" );
document.write( "3π/2| 3.14...\r\n" );
document.write( "5π/3| 3.14...\r\n" );
document.write( "  2π| 3.14...\r\n" );
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document.write( "Which means to swing a radius of 3.14... around from the\r\n" );
document.write( "\"east\" in the counter-clockwise direction through an angle\r\n" );
document.write( "of θ and place a point.  When you do you get this black\r\n" );
document.write( "circle with a radius of 3.14...  Beside it is the same circle\r\n" );
document.write( "plotted in rectangular coordinates.\r\n" );
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document.write( "To get the equation in rectangular coordinates we draw the triangle,\r\n" );
document.write( "and the facts from trigonometry about the sides and hypotenuse of\r\n" );
document.write( "a right triangle that we use to substitute:\r\n" );
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document.write( "So in the equation\r\n" );
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document.write( "1) r = π\r\n" );
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document.write( "we substitute and get\r\n" );
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document.write( "1) \r\n" );
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document.write( "then square both sides and get\r\n" );
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document.write( "1) \r\n" );
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document.write( "And that fits perfectly with what we've learned about \r\n" );
document.write( "equations of circles with center at the origin. \r\n" );
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document.write( "2) θ = 6\r\n" );
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document.write( "Draw an angle of 6 radians (about 344°) measured \r\n" );
document.write( "counter-clockwise from the \"east\" (indicated by the\r\n" );
document.write( "red arc) through the pole (origin), and draw a line.\r\n" );
document.write( "Beside it, we draw the same line in rectangular coordinates.\r\n" );
document.write( "[The red arc is not part of the graph. I just drew it\r\n" );
document.write( "there so you would see the angle that we swing through\r\n" );
document.write( "from the \"east\". On the right graph I also drew in a green\r\n" );
document.write( "perpendicular to the x-axis, so you cold see the right\r\n" );
document.write( "triangle and realize that the slope of the line in rectangular\r\n" );
document.write( "coordinates was y/x which is the tangent of the angle of \r\n" );
document.write( "6 radians]:\r\n" );
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document.write( "The slope of that line is the tangent of 6 radians, and the\r\n" );
document.write( "y-intercept is (0,0), so its equation is found by substituting\r\n" );
document.write( "tan(6) for m and 0 for b in\r\n" );
document.write( "y = mx + b\r\n" );
document.write( "y = tan(6)x + 0\r\n" );
document.write( "y = tan(6)x\r\n" );
document.write( "Using a calculator to get the approximation\r\n" );
document.write( "y = -.2910061914x \r\n" );
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document.write( "3) r = 4/(3cosθ -sinθ )\r\n" );
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document.write( "We could plot a lot of points in polar coordinates and draw\r\n" );
document.write( "it, but it's fairly complicated, so I won't bother on this and\r\n" );
document.write( "the last one.\r\n" );
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document.write( "Use this to make substitutions:\r\n" );
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document.write( "Use this to make substitutions:\r\n" );
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document.write( "Replace cos(θ) by x/r and sin(θ) by y/r\r\n" );
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document.write( "Simplify the right side by multiplying top and bottom by r:\r\n" );
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document.write( "Simplify:\r\n" );
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document.write( "Cross multiply:\r\n" );
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document.write( "Divide both sides by r\r\n" );
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document.write( "4) r^2 = 36/(9-13sin^2(θ) )\r\n" );
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document.write( "Use this to make substitutions:\r\n" );
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document.write( "4)  \r\n" );
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document.write( "Simplify the fraction by multiplying top and bottom by r²\r\n" );
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document.write( "Multiply both sides by the denominator on the right:\r\n" );
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document.write( "Divide both sides by r²\r\n" );
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document.write( "Substitute  r² = x²+y²\r\n" );
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document.write( "     \r\n" );
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document.write( "That will be a hyperbola.\r\n" );
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document.write( "Edwin
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