Stitz-Zeager_College_Algebra_e-book

573 inductive step 573 inection point 386 integers 51

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Unformatted text preview: to use the exponential function. For real numbers t, Euler’s Formula defines eit = cos(t) + i sin(t). (a) Use Theorem 11.16 to show that eix eiy = ei(x+y) for all real numbers x and y . n (b) Use Theorem 11.16 to show that eix = ei(nx) for any real number x and any natural number n. eix (c) Use Theorem 11.16 to show that iy = ei(x−y) for all real numbers x and y . e (d) If z = rcis(θ) is the polar form of z , show that z = reit where θ = t radians. (e) Show that eiπ + 1 = 0. (This famous equation relates the five most important constants in all of Mathematics with the three most fundamental operations in Mathematics.) (f) Show that cos(t) = eit + e−it eit − e−it and that sin(t) = for all real numbers t. 2 2i 11.7 Polar Form of Complex Numbers 11.7.2 857 Answers √ √ 1. (a) z = 9 + 9i = 9 2cis π , Re(z ) = 9, Im(z ) = 9, |z | = 9 2 4 arg(z ) = π + 2πk : k is an integer and Arg(z ) = π . 4 4 (b) z = −5i = 5cis − π , Re(z ) = 0, Im(z ) = −5, |z | = 5 2 arg(z ) = − π + 2πk : k is an integer and Arg(z ) = − π . 2 2 √ √ π 1 (c) z = − 23 − 1 i = cis − 56 , Re(z ) = − 23 , Im(z ) = − 2 , |z | = 1 2 5π π arg(z ) = − 6 + 2πk : k is an integer and Arg(z ) = − 56 . (d) z = −7 + 24i = 25cis π − arctan 274 , Re(z ) = −7, Im(z ) = 24, |z | = 25 arg(z ) = π − arctan 274 + 2πk : k is an integer and Arg(z ) = π − arctan 274 . √ √ √ π 2. (a) z = 12cis − π = 6 − 6i 3 (c) z = 2cis 78 = − 2 + 2 + i 2 ...
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This note was uploaded on 05/03/2013 for the course MATH Algebra taught by Professor Wong during the Fall '13 term at Chicago Academy High School.

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