C10.4 Notes and Formulas. (a) (b) (c) Figure 2 (a) A graph is symmetric with respect to the line θ =

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1 C10.4 Notes and Formulas symmetry tests A polar equation describes a curve on the polar grid. The graph of a polar equation can be evaluated for three types of symmetry, as shown in Figure. Figure A graph is symmetric with respect to the line = _ (y-axis) if replacing (r, ) with ( r, ) yields an equivalent equation. A graph is symmetric with respect to the polar axis (x-axis) if replacing (r, ) with (r, ) or ( r, ) yields an equivalent equation. A graph is symmetric with respect to the pole (origin) if replacing (r, ) with ( r, ) yields an equivalent equation.

2 formulas for the equation of a circle Some of the formulas that produce the graph of a circle in polar coordinates are given by r = acos and r = asin, where a is the diameter of the circle or the distance from the pole to the farthest point on the circumference. The radius is _ a, or one-half the diameter. For r = acos, the center is _ ( a Figure 5 shows the graphs of these four circles., 0 ). For r = asin, the center is ( a _, ). r = acos, a > 0 r = acos, a < 0 r = asin, a > 0 r = asin, a < 0 Figure 5

3 formulas for a cardioid The formulas that produce the graphs of a cardioid are given by r = a ± bcos and r = a ± bsin where a > 0, b > 0, and a = 1. The cardioid graph passes through the pole, as we can see in Figure 7. b r = a + bcos r = a bcos r = a + bsin r = a bsin Figure 7 How To Given the polar equation of a cardioid, sketch its graph. 1. Check equation for the three types of symmetry.. Find the zeros. Set r = Find the maximum value of the equation according to the maximum value of the trigonometric expression. 4. Make a table of values for r and. 5. Plot the points and sketch the graph.

4 b b formulas for one-loop limaçons The formulas that produce the graph of a dimpled one-loop limaçon are given by r = a ± bcos and r = a ± bsin where a > 0, b > 0, and 1 < a <. All four graphs are shown in Figure 9. b r = a + bcos r = a bcos r = a + bsin r = a bsin Figure 9 Dimpled limaçons How To Given a polar equation for a one-loop limaçon, sketch the graph. 1. Test the equation for symmetry. Remember that failing a symmetry test does not mean that the shape will not exhibit symmetry. Often the symmetry may reveal itself when the points are plotted.. Find the zeros. 3. Find the maximum values according to the trigonometric expression. 4. Make a table. 5. Plot the points and sketch the graph.

5 Pascal( ), rediscovered it. formulas for inner-loop limaçons The formulas that generate the inner-loop limaçons are given by r = a ± bcos and r = a ± bsin where a > 0, b > 0, and a < b. The graph of the inner-loop limaçon passes through (7,pole 0) twice: once for the outer loop, and ( 3, ) the once for the inner loop. See Figure 11 for the graphs. Figure 1 Inner-loop limaçon r = a + bcos, a < b r = a bcos, a < b r = a + bsin, a < b r = a bsin, a < b Investigating Lemniscates The lemniscate is a polar curve resembling the infinity symbol or a figure 8. Centered at the pole, a lemniscate is Figure 11 symmetrical by definition. formulas for lemniscates The formulas that generate the graph of a lemniscate are given by r = a cos and r = a sin where a 0. The formula r = a sin is symmetric with respect to the pole. The formula r = a cos is symmetric with respect, and the polar axis. See Figure 13 for the graphs. to the pole, the line = r = a cos() r = a cos() r = a sin() Figure 13 Example 7 Sketching the Graph of a Lemniscate Sketch the graph of r = 4cos. r = a sin()

6 SECTION 10.4 POLAR COORDINATES: GRAPHS 809 rose curves The formulas that generate the graph of a rose curve are given by r = acos n and r = asin n where a 0. If n is even, the curve has n petals. If n is odd, the curve has n petals. See Figure 15. SECTION 10.4 POLAR COORDINATES: GRAPHS Try It #5 Sketch the graph of r = 3cos(3). Investigating the Archimedes Spiralr = acos(n), n even r = asin(n), n odd The final polar equation we will discuss is the Archimedes spiral, named for its discoverer, the Greek mathematician Figure 15 Archimedes (c. 87 BCE c. 1 BCE), who is credited with numerous discoveries in the fields of geometry and mechanics. Example 8 spiral Sketching the Graph of a Rose Curve (n Even) Archimedes Sketch the graph r = cos the 4. graph of the Archimedes spiral is given by r = for 0. As increases, r increases The formula thatofgenerates at a constant rate in an ever-widening, never-ending, spiraling path. See Figure 18. Solution Testing for symmetry, we find again that the symmetry tests do not tell the whole story. The graph is not only symmetric with respect to the polar axis, but also with respect to the line = _ and the pole. Now we will find the zeros. First make the substitution u = 4. 0 = cos 4 0 = cos 4 0 = cos u cos 1 0 = u r =, [0, ] How To u= _ Figure 18 _ 4 = _ = 8 r =, [0, 4] 811

7 Summary of Curves We have explored a number of seemingly complex polar curves in this section. Figure 0 and Figure 1 summarize the graphs and equations for each of these curves. Circle Cardioid One-Loop Limaçon Inner-Loop Limaçon r = asin r = acos r = a ± bcos r = a ± bsin a > 0, b > 0, a/b = 1 r = a ± bcos r = a ± bsin a > 0, b > 0, 1 < a/b < r = a ± bcos r = a ± bsin a > 0, b > 0, a < b Figure 0 Lemniscate Rose Curve (n even) Rose Curve (n odd) Archimedes Spiral r = a cos r = a sin a 0 r = acos n r = asin n n even, n petals r = acos n r = asin n n odd, n petals r = 0 Figure 1