6. (a) Let a be a real number in the interval [0, 27), and consider the complex number cos(a) + i sin(a). Prove that 2" = cos(na) + i sin(na) for all n e N. (b) Let k, n E N, and consider the complex number u = cos(2Tk/n) +i sin(27k/n). Prove that u is an n-th root of unity (i.e. u" = 1). (c) Find all 6 (complex) eigenvalues of the matrix 0 0 0 0 0 1 1 0 0 0 0 0 0 10 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0

Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter7: Analytic Trigonometry
Section7.3: The Addition And Subtraction Formulas
Problem 79E
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6. (a) Let a be a real number in the interval [0, 27), and consider the
complex number
cos(a) + i sin(a).
Prove that 2" = cos(na) + i sin(na) for all n e N.
(b) Let k, n E N, and consider the complex number
u = cos(2Tk/n) +i sin(27k/n).
Prove that u is an n-th root of unity (i.e. u" = 1).
(c) Find all 6 (complex) eigenvalues of the matrix
0 0 0 0 0 1
1 0 0 0 0 0
0 10 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
Transcribed Image Text:6. (a) Let a be a real number in the interval [0, 27), and consider the complex number cos(a) + i sin(a). Prove that 2" = cos(na) + i sin(na) for all n e N. (b) Let k, n E N, and consider the complex number u = cos(2Tk/n) +i sin(27k/n). Prove that u is an n-th root of unity (i.e. u" = 1). (c) Find all 6 (complex) eigenvalues of the matrix 0 0 0 0 0 1 1 0 0 0 0 0 0 10 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0
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