610 CHAPTER 9 Polar Coordinates; Vectors 9.2 Polar Equations and Graphs Now Work the ‘Are You Prepared?’ problems on page 624. • Symmetry (Section 1.3, pp. 21–23) • Circles ( Section 1.6 , pp. 48 – 52 ) • Even–Odd Properties of Trigonometric Functions (Section 6.3, pp. 422–423) • Difference Formulas for Sine and Cosine (Section 7.5, pp. 511 and 514) • Values of the Sine and Cosine Functions at Certain Angles (Section 6.2, pp. 399–405) PREPARING FOR THIS SECTION Before getting started, review the following: Figure 22 O u 5 0 u 5 p u 5 u 5 u 5 u 5 u 5 u 5 p– 2 3p –– 2 7p –– 4 p– 4 3p –– 4 5p –– 4 x 24 2221 23 4 1 2 3 O y 2 4 24 1 23 3 22 (1, 2) (23, 1) (a) Rectangular grid r 5 5 r 5 3 r 5 1 2, 4, ( ) ( ) (b) Polar grid p– 4 5p –– 4 DEFINITION Polar Equation An equation whose variables are polar coordinates is called a polar equation . The graph of a polar equation consists of all points whose polar coordinates satisfy the equation. OBJECTIVES 1 Identify and Graph Polar Equations by Converting to Rectangular Equations (p. 611) 2 Graph Polar Equations Using a Graphing Utility (p. 612) 3 Test Polar Equations for Symmetry (p. 615) 4 Graph Polar Equations by Plotting Points (p. 616) Just as a rectangular grid may be used to plot points given by rectangular coordinates, such as the points 3, 1 ( ) − and 1, 2 ( ) shown in Figure 22(a), a grid consisting of concentric circles (with centers at the pole) and rays (with vertices at the pole) can be used to plot points given by polar coordinates, such as the points 4, 5 4 π ( ) and 2, 4 π ( ) shown in Figure 22(b). Such polar grids are used to graph polar equations . 1. x y 22 2 22 2 4 (3, 21) ; quadrant IV 2. ( ) ( ) − + − x x y y 2 1 2 2 1 2 3. 9 4. x y 5p — 6 5. b a 6. π − 4 ‘Are You Prepared?’ Answers 98. If = − z i 2 5 and = + w i 4 , find ⋅ z w. 99. Solve the equation: θ θ θ π = ≤ < 4 sin cos 1, 0 2 100. Solve the triangle: = ° = ° = A B c 65 , 37 , 10 101. Find the exact value of sin π7 12 . 102. Simplify: ( ) ⋅ − ⋅ x e e x x 5 3 10 5 x x 2 3 3 2 2 103. Show that = − + x x x x x x sin sin 2 cos sin cos sin 5 2 4 .
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