616 CHAPTER 9 Polar Coordinates; Vectors Figure 30 (a) Points symmetric with respect to the polar axis x u 5 0 u 5 p u 5 p– 2 u 5 3p –– 2 u 5 7p –– 4 u 5 p– 4 u 5 3p –– 4 u 5 5p –– 4 1 2 3 4 5 O y u (r, u) (r, 2u) 5 (2r, p 2 u) p–– 2 (b) Points symmetric with respect to the line u 5 x u 5 0 u 5 p p– 2 3p –– 2 7p –– 4 p– 4 u 5 u 5 u 5 u 5 u 5 3p –– 4 u 5 5p –– 4 1 2 3 4 5 O y u u (r, u) u (r, p 2 u) 5 (2r, 2u) 5 (r, u 1 p) u 1 p (c) Points symmetric with respect to the pole x u 5 0 u 5 p u 5 p– 2 u 5 3p –– 2 u 5 7p –– 4 u 5 p– 4 u 5 3p –– 4 u 5 5p –– 4 1 2 3 4 5 y O u (2r, u) (r, u) 1 2 3 4 p 2 u p 2 u 2u 2u The following tests are a consequence of these observations. THEOREM Tests for Symmetry • Symmetry with Respect to the Polar Axis (x-Axis) In a polar equation, replace θ by ,θ− or replace r by r− and θ by . π θ − If an equivalent equation results from either substitution, the graph is symmetric with respect to the polar axis. • Symmetry with Respect to the Line 2 θ π = (y-Axis) In a polar equation, replace θ by , π θ − or replace r by r− and θ by .θ− If an equivalent equation results from either substitution, the graph is symmetric with respect to the line 2 . θ π = • Symmetry with Respect to the Pole (Origin) In a polar equation, replace r by r, − or replace θ by . θ π + If an equivalent equation results from either substitution, the graph is symmetric with respect to the pole. The tests for symmetry are sufficient conditions for symmetry, but they are not necessary conditions. That is, an equation may fail these tests and still have a graph that is symmetric with respect to the polar axis, the line 2 , θ π = or the pole. 4 Graph Polar Equations by Plotting Points Graphing a Polar Equation (Cardioid) Graph the equation: r 1 sinθ = − Solution EXAMPLE 8 Check for symmetry first. Polar Axis: Replace θ by .θ− The result is r 1 sin 1 sin θ θ ( ) = − − = + sin sin θ θ ( ) − =− The test fails. Replace r by r− and θ by . π θ − The result is r r r r r 1 sin 1 sin cos cos sin 1 0 cos 1 sin 1 sin 1 sin π θ π θ π θ θ θ θ θ [ ] ( ) ( ) [ ] − = − − − = − − − = − ⋅ − − − = − = − + This test also fails, so the graph may or may not be symmetric with respect to the polar axis.
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