# The curves sketched in examples 6 and 8 are symmetric

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The curves sketched in Examples 6 and 8 are symmetric about the polar axis, since. The curves in Examples 7 and 8 are symmetric aboutbecauseand. The four-leaved rose is also symmetricabout the pole. These symmetry properties could have been used in sketching the curves.For instance, in Example 6 we need only have plotted points forand thenreflected about the polar axis to obtain the complete circle.Tangents to Polar CurvesTo find a tangent line to a polar curve, we regardas a parameter and write itsparametric equations asThen, using the method for finding slopes of parametric curves (Equation 10.2.1) and theProduct Rule, we haveWe locate horizontal tangents by finding the points where(provided that). Likewise, we locate vertical tangents at the points where(pro-vided that).Notice that if we are looking for tangent lines at the pole, thenand Equation 3 sim-plifies torr2O(r, ¨)(_r, ¨)O(r, ¨)(r, _¨)¨(a)(b)(c)FIGURE 14O(r, ¨)(r, π-¨)π-¨¨2coscoscos 2cos 2sinsin02rfyrsinfsinxrcosfcosdydxdyddxddrdsinrcosdrdcosrsin3dy d0dx d0dx d0dy d0r0drd0ifdydxtan
684CHAPTER 10For instance, in Example 8 we found thatwhenor. Thismeans that the linesand(orand) are tangent lines toat the origin.(a) For the cardioidof Example 7, find the slope of the tangent linewhen(b) Find the points on the cardioid where the tangent line is horizontal or vertical.rcos 20434434yxyxrcos 2EXAMPLE 9.andarecossin2,,12ππ2 ¨
POLAR COORDINATES685NOTEInstead of having to remember Equation 3, we could employ the method used toderive it. For instance, in Example 9 we could have writtenThen we havewhich is equivalent to our previous expression.

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Term
Fall
Professor
John
Tags
Parametric equation, Conic section
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