 8.1: Write 12 in terms of i.
 8.2: Find x and y so that the equation (x2 3x) 16i 10 8yi is true. 3
 8.3: Simplify (6 3i) [(4 2i) (3 i)].
 8.4: Simplify i 17.
 8.5: (8 5i)(8 5i)
 8.6: (3 5i) 2
 8.7: Divide . Write your answer in standard form.
 8.8: 3 4i
 8.9: 8i
 8.10: 8(cos 330 i sin 330)
 8.11: 2 cis
 8.12: 3 i 13.
 8.13: 5i
 8.14: 5(cos 25 i sin 25) 3(cos 40 i sin 40)
 8.15: 5(cos 25 i sin 25) 3(cos 40 i sin 40)
 8.16: [3 cis 20]4
 8.17: Find two square roots of z 49(cos 50 isin 50). Leave your answer in...
 8.18: Find the 4 fourth roots ofz 2 2i3. Leave your answer in trigonometr...
 8.19: x4 23x2 4 0 20.
 8.20: x3 1
 8.21: Convert the point (6, 60) to rectangular coordinates, and state two...
 8.22: Convert (3, 3) to polar coordinates with r positive and between 0 a...
 8.23: Convert the equation r 6 sin to rectangular coordinates.
 8.24: Convert the equation x2 y2 8y to polar coordinates.
 8.25: r 4 2
 8.26: r 4 2 cos 27.
 8.27: r sin 2
 8.28: r 4 4 cos 29
 8.29: r 6 cos 330
 8.30: 3 sin 2sin G
Solutions for Chapter 8: Complex Numbers and Polar Coordinates
Full solutions for Trigonometry  7th Edition
ISBN: 9781111826857
Solutions for Chapter 8: Complex Numbers and Polar Coordinates
Get Full SolutionsChapter 8: Complex Numbers and Polar Coordinates includes 30 full stepbystep solutions. Trigonometry was written by and is associated to the ISBN: 9781111826857. Since 30 problems in chapter 8: Complex Numbers and Polar Coordinates have been answered, more than 24711 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Trigonometry, edition: 7. This expansive textbook survival guide covers the following chapters and their solutions.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

Iterative method.
A sequence of steps intended to approach the desired solution.

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Solvable system Ax = b.
The right side b is in the column space of A.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.