# Y y x 4 2 2 0 4 2 2 y y x 2 2 2 2 2 y x square the

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y y x 4 2 2 + 0 4 2 2 - + y y x 2 2 2 2 2 - + y x square the complete we Here, This is the circle of radius 2 with center ( x , y ) ( 0, 2 ). Example Describe the graphs of the following polar equations and find the corresponding rectangular equation. q p q sec (c) 3 (b) 2 (a) r r
15 Solution (a) The graph of the polar equation r 2 consists of all points that are two units from the pole. 0 polar axis 2 p p 2 3 p 2 Circle, r 2. 16 get to ip relationsh the use we convert, To 2 2 2 y x r + 2 r 2 2 2 r 2 2 2 2 + y x Polar equation Rectangular equation (b) 3 p q . 3 of angle an makes that LINE on the points all of consists 3 equation polar the of graph The p p q 2 p p 2 3 p polar axis 3 p q radial line 0
17 get to tan ip relationsh the use we convert, To x y q 3 p q 3 tan q x y x y 3 or 3 Polar equation Rectangular equation (c) . cos using by form r rectangula o convert t we so , inspection simple by evident so not is sec of graph The q q r x r q sec r q cos 1 r 1 or cos 1 x r q Polar equation Rectangular equation Now, we see that the graph is a vertical line. 18 2 p p 2 3 p polar axis 1 q sec line, Vertical r 0
19 If a polar curve is symmetric, then we can use it to our advantage. Let us look at a symmetric polar curve, called the lemniscate , produced by r = f ( q ). Observation 1 : Symmetricity about the x -axis. The sign and size of r do not change even if we use q or - q . That is, r = f ( q ) = f ( -q ). q - q 20 Observation 2 : Symmetricity about the y -axis. The sign and size of r do not change even if we use q or p-q . That is, r = f ( q )= f ( p-q ). q p - q
21 2 Observation 3 : Symmetricity about the origin . q p + q The sign and size of r do not change even if we use q or p+ q . That is, r = f ( q ) = f ( p + q ). 22 Symmetric Polar Curves Let us look at a symmetric polar curve, called the lemniscate , produced by r = f ( q ). Symmetricity about the x -axis. The sign and size of r do not change even if we use q or - q . That is, r = f ( q ) = f ( -q ). q - q 6 p 5 6 p 7 6 p
23 Symmetricity about the y -axis. The sign and size of r do not change even if we use q or p-q . That is, r = f ( q )= f ( p-q ). q p - q 6 p 5 6 p 7 6 p 24 2 Symmetricity about the origin . q p + q The sign and size of r do not change even if we use q or p + q . That is, r = f ( q ) = f ( p + q ). 6 p 5 6 p 7 6 p
25 Tests for Symmetry  Symmetricity about the polar axis (or x -axis)  axis) - (or 2 line about the ty Symmetrici y p q  Symmetricity about the pole axis (or origin) We get back if we replace by . r f q q p q - We get back if we replace by . r f q q q - We get back r = f ( q ) if we replace q by p + q . 26 Example . cos 2 3 sketch plot to - point and symmetry Use q + r Solution Test for Symmetry about the x -axis Replace by to get q q - Thus 3 2cos is symmetric about the -axis. r x q + r + + + + + - + - + q q q q q q q cos 2 3 sin 0 cos 1 2 3 sin 0 sin cos 0 cos 2 3 0 cos 2 3 cos 2 3
27 Test for Symmetry about the y -axis Thus 3 2cos is not symmetric about the -axis. r y q + get to by Replace q p q - r - + - + + + - + q q q q p q p q p cos 2 3 sin 0 cos 1 2 3 sin sin cos cos 2 3 cos 2 3 28 Test for Symmetry about the origin Thus 3 2cos is not symmetric about the origin. r q + get to by Replace q p q + r - - - + - + + + q q q q p q p q p cos
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