# Section 10.2 Graphing Polar Equations

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1 Section 10.2 Grphing Polr Equtions OBJECTIVE 1: Sketching Equtions of the Form rcos, rsin, r cos r sin c nd Grphs of Polr Equtions of the Form rcos, rsin, r cos r sin c, nd where,, nd c re constnts. The grph of rcos is verticl line. The grph of rsin is horizontl line. The grph of rcos r sin c is line with slope m nd y intercept. The grph of is line through the pole tht mkes n ngle of with the polr xis.

2 Sketch the grph of the polr eqution

3 OBJECTIVE 2: Sketching Equtions of the Form r, r sin, nd r cos nd r cos Grphs of Polr Equtions of the Form r, r sin, nd cos r where 0 is constnt The grph of r is circle The grph of r sin is circle The grph of r cos is circle centered t the pole with centered on the line = tht is units rdius of length. centered on the polr xis tht is units 2 from the pole with rdius of length. from the pole with rdius of length.

4 Sketch the grph of the polr eqution

5 OBJECTIVE 3: Sketching Equtions of the Form r sin nd r cos Grphs of Polr Equtions of the Form r sinnd r cos where 0 nd 0 re constnts. The grph is crdioid if 1. The grph is limcon with n inner loop if <1. The grph is limcon with dimple if 1< <2. The grph is limcon with no inner loop nd no dimple if 2.

6 Steps for Sketching Polr Equtions (Limcons) of the Form r sinnd r cos Step 1. Identify the generl shpe using the rtio. If 1, then the grph is crdioid. If 1, then the grph is limcon with n inner loop tht intersects the pole. If 1 2, then the grph is limcon with dimple.. If 2, then the grph is limcon with no inner loop nd no dimple. Step 2. Determine the symmetry. If the eqution is of the form If the eqution is of the form r sin, then the grph must e symmetric out the line. 2 r cos, then the grph must e symmetric out the polr xis. Step 3. Plot the points corresponding to the qudrntl ngles 0,,, nd Step 4. If necessry, plot few more points until symmetry cn e used to complete the grph Sketch the grph of the eqution.

7 OBJECTIVE 4: Sketching Equtions of the Form r sin n nd r cos n Grphs of Polr Equtions of the Form r sin n nd r cos n where 0 is constnt nd n 1 is positive integer. The grph is rose with n petls. The endpoint of one petl lies 3 long the verticl line = or =. The grph is rose with n petls. The endpoint of one petl lies long the polr xis. The grph is rose with 2 n petls. None of the petls hve n endpoint lying on either the polr xis or the line =. 2 The grph is rose with 2 n petls. 3 on the lines = nd =. Two petls hve endpoints tht lie on the polr xis.

8 Steps for Sketching Polr Equtions (Roses) of the Form Step 1. Identify the numer of petls. If n is even, then there re 2n petls. If n is odd, then there re n petls. r sin n nd r cos n where 0 nd n 1is positive integer. Step 2. Determine the length of ech petl. The length of ech petl is units. Step 3. Determine ll ngles where n endpoint of petl lies. If the eqution is of the form r sin n sin n 1 ndsin n 1. If the eqution is of the form r cos n cos n 1 nd cos n 1., then the endpoints occur for ngles on the intervl, then the endpoints occur for ngles on the intervl 0,2 tht stisfy the equtions 0,2 tht stisfy the equtions Step 4. Plot ll points corresponding to the vlues of found in Step 3. These points represent the endpoints of ech petl. Step 5. Determine ngles where the grph psses through the pole. These ngles serve s guide when sketching the width of petl. If the eqution is of the form r sin n, then the grph psses through the pole when sin n 0. If the eqution is of the form r cos n, then the grph psses through the pole when cos n 0. Step 6. Drw ech petl to complete the grph.

9 Grph. OBJECTIVE 5: Sketching Equtions of the Form r sin2 nd r cos2 Grphs of Polr Equtions of the Form r 2 2 sin 2 nd r cos2 where 0 is constnt The grph is lemniscte symmetric out the pole with the endpoints of the two loops lying 5 long the ngles nd. 4 4 The grph is lemniscte symmetric out the pole, the polr xis, nd the verticl line =. The endpoints of the two loops lie long the 2 ngles 0nd.

10 Grph.

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