Chapter Review 675 Chapter Review Things to Know Polar Coordinates (pp. 600–607) Relationship between polar coordinates θ ( ) r, and rectangular coordinates ( ) x y , (pp. 602 and 606) θ θ θ θ π = = = + = ≠ = = = x r y r r x y y x x r y x cos , sin , tan , if 0 , 2 , if 0 2 2 2 Complex Numbers and De Moivre’s Theorem (pp. 627–634) Polar form of a complex number (p. 628) If = + z x yi, then θ θ ( ) = + z r i cos sin , where θ π ≥ ≤ < r 0 and 0 2 . Exponential form of a complex number (p. 629) = θ z re , i where θ θ θ = + θ e i cos sin , i in radians De Moivre’s Theorem (p. 631) If = θ z re , i then = θ ( ) z r e , n n i n where ≥ n 1 is an integer. nth root of a complex number = ≠ θ w re w , 0 i (p. 633) … = = − θ π ( )( ) + z r e k n , 0, , 1, k n i n k 1 2 where ≥ n 2 is an integer Vectors (pp. 637–646) A quantity having magnitude and direction; equivalent to a directed line segment PQ Position vector (pp. 640 and 661) A vector whose initial point is at the origin Unit vector (pp. 639, 643 and 663) A vector whose magnitude is 1 Direction angle of a vector v (p. 644) The angle α, α ° ≤ < ° 0 360 , between i and v Dot product (pp. 652 and 663) If = + a b v i j 1 1 and = + a b w i j, 2 2 then ⋅ = + a a b b v w . 1 2 1 2 If = + + = + + a b c a b c v i j k w i j k and 1 1 1 2 2 2 then ⋅ = = + a a b b c c v w . 1 2 1 2 1 2 Angle θ between two nonzero vectors u and v (pp. 653 and 664) θ = ⋅ u v u v cos , θ π ≤ ≤ 0 Work (p. 656) Work ( )( ) = = W AB F magnitude of force distance = ⋅ W AB F Direction angles of a vector in space (p. 665) If = + + a b c v i j k, then α β γ [ ] ( ) ( ) ( ) = + + v v i j k cos cos cos , where α β γ = = = a b c v v v cos , cos , and cos . Cross product (p. 669) If = + + a b c v i j k 1 1 1 and = + + a b c w i j k, 2 2 2 then [ ] [ ] [ ] × = − − − + − b c b c a c a c a b a b v w i j k. 1 2 2 1 1 2 2 1 1 2 2 1 Area of a parallelogram (p. 672) θ × = u v u v sin , where θ is the angle between the two adjacent sides u and v. Objectives Section You should be able to . . . Example(s) Review Exercises 9.1 1 Plot points using polar coordinates (p. 600) 1–3 1–3 2 Convert from polar coordinates to rectangular coordinates (p. 602) 4 1–3 3 Convert from rectangular coordinates to polar coordinates (p. 604) 5–7 4–6 4 Transform equations between polar and rectangular forms (p. 606) 8, 9 7(a)–10(a) 74. Rationalize the numerator: − x x 4 75. Find the average rate of change of ( ) = − f x x csc 1 from 1 to 2. 71. Find the domain of ( ) = + − f x x x 3 4 16 . 2 72. If θ π θ π = − < < cos 3 8 , 2 , find the exact value of θ sin 2 . 73. Find the area of the triangle for which = = a b 8, 9, and = ° C 60 .
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