 8.6.8.1.326: Sketch the graph of each equation by making a table using values of...
 8.6.8.1.327: Sketch the graph of each equation by making a table using values of...
 8.6.8.1.328: Sketch the graph of each equation by making a table using values of...
 8.6.8.1.329: Sketch the graph of each equation by making a table using values of...
 8.6.8.1.330: Use your graphing calculator in polar mode to generate a table for ...
 8.6.8.1.331: Use your graphing calculator in polar mode to generate a table for ...
 8.6.8.1.332: Use your graphing calculator in polar mode to generate a table for ...
 8.6.8.1.333: Use your graphing calculator in polar mode to generate a table for ...
 8.6.8.1.334: Graph each equation. r = 3
 8.6.8.1.335: Graph each equation. r = 2
 8.6.8.1.336: Graph each equation. (J = ..JI
 8.6.8.1.337: Graph each equation. (J = 3'71
 8.6.8.1.338: Graph each equation. r 3 sin (J
 8.6.8.1.339: Graph each equation. r = 3 cos (J
 8.6.8.1.340: Graph each equation. r 4 + 2 sin (J
 8.6.8.1.341: Graph each equation. r = 4 + 2 cos (J
 8.6.8.1.342: Graph each equation. r = 2 + 4 cos (J
 8.6.8.1.343: Graph each equation. r = 2 + 4 sin (J
 8.6.8.1.344: Graph each equation. r = 2 + 2 sin (J
 8.6.8.1.345: Graph each equation. r 2 + 2 cos (J
 8.6.8.1.346: Graph each equation. r2 = 9 sin 2(J
 8.6.8.1.347: Graph each equation. r2 = 4 cos 2(J
 8.6.8.1.348: Graph each equation. r 2 sin 2(J
 8.6.8.1.349: Graph each equation. r = 2 cos 2(J
 8.6.8.1.350: Graph each equation. r = 4 cos 3(J
 8.6.8.1.351: Graph each equation. r 4 sin 3(J
 8.6.8.1.352: Graph each equation using your graphing calculator in polar mode. r...
 8.6.8.1.353: Graph each equation using your graphing calculator in polar mode. r...
 8.6.8.1.354: Graph each equation using your graphing calculator in polar mode. r...
 8.6.8.1.355: Graph each equation using your graphing calculator in polar mode. r...
 8.6.8.1.356: Graph each equation using your graphing calculator in polar mode. r...
 8.6.8.1.357: Graph each equation using your graphing calculator in polar mode. r...
 8.6.8.1.358: Graph each equation using your graphing calculator in polar mode. r...
 8.6.8.1.359: Graph each equation using your graphing calculator in polar mode. r...
 8.6.8.1.360: Graph each equation using your graphing calculator in polar mode. r...
 8.6.8.1.361: Graph each equation using your graphing calculator in polar mode. r...
 8.6.8.1.362: Graph each equation using your graphing calculator in polar mode. r...
 8.6.8.1.363: Graph each equation using your graphing calculator in polar mode. r...
 8.6.8.1.364: Convert each equation to polar coordinates and then sketch the grap...
 8.6.8.1.365: Convert each equation to polar coordinates and then sketch the grap...
 8.6.8.1.366: Convert each equation to polar coordinates and then sketch the grap...
 8.6.8.1.367: Convert each equation to polar coordinates and then sketch the grap...
 8.6.8.1.368: Convert each equation to polar coordinates and then sketch the grap...
 8.6.8.1.369: Convert each equation to polar coordinates and then sketch the grap...
 8.6.8.1.370: Change each equation to rectangular coordinates and then graph. r(2...
 8.6.8.1.371: Change each equation to rectangular coordinates and then graph. r(3...
 8.6.8.1.372: Change each equation to rectangular coordinates and then graph. r( ...
 8.6.8.1.373: Change each equation to rectangular coordinates and then graph. r(1...
 8.6.8.1.374: Change each equation to rectangular coordinates and then graph. r 4...
 8.6.8.1.375: Change each equation to rectangular coordinates and then graph. r =...
 8.6.8.1.376: Graph rl = 2 sin (J and rz = 2 cos (J and then name two points they...
 8.6.8.1.377: Graph rl = 2 + 2 cos (J and r2 = 2 2 cos 8 and name three points th...
 8.6.8.1.378: The problems that follow review material we covered in Section 4.6....
 8.6.8.1.379: The problems that follow review material we covered in Section 4.6....
 8.6.8.1.380: The problems that follow review material we covered in Section 4.6....
 8.6.8.1.381: The problems that follow review material we covered in Section 4.6....
 8.6.8.1.382: The problems that follow review material we covered in Section 4.6....
 8.6.8.1.383: The problems that follow review material we covered in Section 4.6....
Solutions for Chapter 8.6: Complex Numbers and Polar Coordinates
Full solutions for Trigonometry
ISBN: 9780495108351
Solutions for Chapter 8.6: Complex Numbers and Polar Coordinates
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Since 58 problems in chapter 8.6: Complex Numbers and Polar Coordinates have been answered, more than 31792 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Trigonometry, edition: . Trigonometry was written by and is associated to the ISBN: 9780495108351. Chapter 8.6: Complex Numbers and Polar Coordinates includes 58 full stepbystep solutions.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

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 •

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.

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

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·

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).