SECTION 9.3 The Complex Plane; De Moivre’s Theorem 627 Figure 38 Complex plane y Imaginary axis Real axis x z 5 x 1 yi O 9.3 The Complex Plane; De Moivre’s Theorem Now Work the ‘Are You Prepared?’ problems on page 634. • Complex Numbers (Section A.7, pp. A59–A63) • Values of the Sine and Cosine Functions at Certain Angles (Section 6.2, pp. 399–405) • Sum and Difference Formulas for Sine and Cosine (Section 7.5, pp. 511 and 514) • Laws of Exponents (Section A.1, pp. A7–A9) PREPARING FOR THIS SECTION Before getting started, review the following: OBJECTIVES 1 Plot Points in the Complex Plane (p. 627) 2 Convert a Complex Number between Rectangular Form and Polar Form or Exponential Form (p. 628) 3 Find Products and Quotients of Complex Numbers (p. 630) 4 Use De Moivre’s Theorem (p. 631) 5 Find Complex Roots (p. 632) 1 Plot Points in the Complex Plane Complex numbers are discussed in Section A.7. In that discussion, we were not prepared to give a geometric interpretation of a complex number. Now we are ready. A complex number = + z x yi can be interpreted geometrically as the point ( ) x y , in the xy -plane. Each point in the plane corresponds to a complex number, and conversely, each complex number corresponds to a point in the plane.The collection of such points is referred to as the complex plane . The x -axis is referred to as the real axis , because any point that lies on the real axis is of the form = + = z x i x 0 , a real number.The y -axis is called the imaginary axis , because any point that lies on the imaginary axis is of the form = + = z yi yi 0 , a pure imaginary number. See Figure 38. Explaining Concepts 93. Write down two different tests for symmetry with respect to the polar axis. Find examples in which one test works and the other fails. Which test do you prefer to use? Justify your answer. 94. Explain why the vertical-line test used to identify functions in rectangular coordinates does not work for equations expressed in polar coordinates. 95. Solve: x 5 3 1 − ≥ 96. Convert 7 3 π radians to degrees. 97. Determine the amplitude and period of y x 2sin 5( ) = − without graphing. 98. Find any asymptotes for the graph of R x x x x 3 12 2 ( ) = + − − 99. Find the remainder when x x x 3 2 7 5 5 3 − + − is divided by x 1. − 100. Find the area of a triangle with sides 6, 11, and 13. 101. Solve: 3 9 x x 2 3 1 = − − 102. Solve: x x 6 7 20 2 + = 103. m f x x x 3 8 2 ( ) = ′ = + gives the slope of the tangent line to the graph of f x x x4 5 3 2 ( ) = + − at any number x. Find an equation of the tangent line to f at x 2. = − 104. Show that x x x x cos cos sin cos . 3 2 = − Retain Your Knowledge Problems 95–104 are based on previously learned material.The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for subsequent sections, a final exam, or later courses such as calculus. ‘Are You Prepared?’ Answers 1. ( ) −4, 6 2. + A B A B cos cos sin sin 3. ( ) ( ) + + − = x y 2 5 9 2 2 4. Odd 5. − 2 2 6. − 1 2
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