SECTION 9.3 The Complex Plane; De Moivre’s Theorem 629 Also, because ≥ r 0, we have = + r x y . 2 2 From equation (1), it follows that the magnitude of z r i cos sin θ θ ( ) = + is Need to Review? The number e is defined in Section 5.3, pp. 302–303. Writing a Complex Number in Polar Form and in Exponential Form Write = − z i 3 in polar form and in exponential form. Solution EXAMPLE 2 Because = x 3 and = − y 1, it follows that ( ) ( ) = + = + − = = r x y 3 1 4 2 2 2 2 2 so θ θ θ π = = = = − ≤ < x r y r cos 3 2 sin 1 2 0 2 The angle θ θ π ≤ < , 0 2 , that satisfies both equations is θ π = 11 6 . With θ π = 11 6 and = r 2, the polar form of = − z i 3 is z r i i cos sin 2 cos 11 6 sin 11 6 θ θ π π ( ) ( ) = + = + The exponential form of = − z i 3 is = = θ π ⋅ z re e2 i i 11 /6 θ π = = r 11 6 , 2 Plotting a Point in the Complex Plane and Converting It to Rectangular Form Plot the point corresponding to π π ( ) = + = π⋅ z i e 2 cos 6 sin 6 2 i 6 in the complex plane, and convert z to rectangular form. EXAMPLE 3 = z r THEOREM Euler’s Formula For any real number θ, e i cos sin i θ θ = + θ r i re cos sin i θ θ ( ) + = θ (4) Leonhard Euler (1707–1783) established a relationship, known as Euler’s Formula , between complex numbers and the number e. The proof of Euler’s Formula requires mathematics beyond the level of this text, so it is not included here. Euler’s Formula allows us to write the polar form of a complex number using exponential notation. When a complex number is written in the form = θ z re , i it is said to be written in exponential form . Note in Euler’s Formula that θ is a real number. That is, θ is in radians. Now Work PROBLEM 13 (continued)
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