Note that in the rectangular system there is only one way to label a point In

# Note that in the rectangular system there is only one

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Note that in the rectangular system, there is only one way to label a point. In the polar system, there are several ways to label a point, actually an infinite number of ways. Example 1) For each polar point, label it in two other ways: a. 4,60°()b. "5,315°()c. 2,"90°()d. 1,5"6e. "8,#6f. "32,"5#3"4,240°(), 4,"300°()5,135°(), 5,"45°()2,270°(),"2,90°()#\$%&(\$%&()\$%&()"1,11#6\$%&(), 1,"7#6\$%&()8,7"6#\$%&(, 8,)5"6#\$%&(32,4"3#\$%&(,32,)2"3#\$%&(To convert to and from the polar system to the coordinate system, you must know the following relationships. xy!rx=rcos!tan!=yxy=rsin!r2=x2+y2r,!()or x,y()Example 2) Convert the following polar points to rectangular coordinates. a. 6,90°()b. 4,60°()c. 10,225°()0,6()2,23()"52,"52()d. (5, π) e. 23,"6#\$%&(f. 52,5"3#\$%&("5,0()3,3()52."532#\$%&(Example 3) Convert the following rectangular points to polar coordinates. a. (-5, -5) b. (0, -2) c. 1,"3()5,225°()2,270°()2,300°()d. "7,0()e. 5,12()f. 6,"3()7,180°()13,67.38°()3 5,43°()
10. Parametric and Polar Equations - 7 - - Stu SchwartzThe TI-84 calculator is capable of making these conversions although it is slightly cumbersome. The commands are located in the ANGLE menus. To convert the rectangular point 4,43()to polar form, you use 5 :R"Pr(and 6 :R"P#.(These commands ask for the value of rand the value of "as two separate statements. The value of "will depend on whether you are in degree or radian mode. To convert the polar point 22,225°()to rectangular form, you use 7 :P"Rx( and 8 :P"Ry(.These commands ask for the values of xand the value of yas two separate statements. Again, when you input ", be sure it matches the form you specified in Mode – radian or degree. Example 4) Convert the following rectangular equations to polar equations. Confirm by calculator. a) x2+y2=25b) x+2()2+y2=4c. y=3r2=25r=5x2+4x+4+y2=4r2+4rcos"=0r r+4cos"()=0r=#4cos"rsin"=3r=3sin"d. x=3e. xy=1f. 2x"3y"2=0r=3cos"rcos"#rsin"=1r2=1sin"cos"r=1sin"cos"2rcos"#3rsin"=2r2cos"#3sin"()=2r=22cos"#3sin"Example 5) Convert the following polar equations to rectangular equations. Confirm by calculator. a) r= 2 b) "=2#3c. r=4sec"x2+y2=2x2+y2=4tan"=#3yx=#3y=#x3r=4cos"rcos"=4x=4
10. Parametric and Polar Equations - 8 - - Stu Schwartzd) r="2csc#e) r=123sin"#4cos"f) r=31+sin"r="2sin#rsin#="2y="23rsin"#4rcos"=123y#4x=12y=4x+123r+y=3x2+y2+y=3x2+y2=3"yx2+y2=9"6y+y2y=9"x26Graphs of Polar EquationsExample 6) Plot the points and sketch the graph of the polar equation r=3cos". (1 decimal place) "0°30°60°90°120°150°180°210°240°270°300°330°360°r 3.0 2.6 1.5 0.0 -1.5 -2.6 -3.0 -2.6 -1.5 0.0 1.5 2.6 3.0 You can graph polar equations on your calculators. You must switch to polar mode using your MODE button as shown to your right. You can graph in either degree or radian mode although you will gain more control with degree mode. However, there are graphs that MUST be graphed in radian mode. Any polar equation using "not involving trig functions must be in radian mode.

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