 8.2.1: Concept Check The absolute value (or modulus) of a complex number r...
 8.2.2: Concept Check What is the geometric interpretation of the argument ...
 8.2.3: Graph each complex number. See Example 1. 3 + 2i
 8.2.4: Graph each complex number. See Example 1. 6  5i
 8.2.5: Graph each complex number. See Example 1. 22 + 22i
 8.2.6: Graph each complex number. See Example 1. 2  2i23
 8.2.7: Graph each complex number. See Example 1. 4i
 8.2.8: Graph each complex number. See Example 1. 3i
 8.2.9: Graph each complex number. See Example 1. 8 1
 8.2.10: Graph each complex number. See Example 1. 2 1
 8.2.11: Concept Check Give the rectangular form of the complex number shown.
 8.2.12: Concept Check Give the rectangular form of the complex number shown.
 8.2.13: Find the sum of each pair of complex numbers. In Exercises 1316, gr...
 8.2.14: Find the sum of each pair of complex numbers. In Exercises 1316, gr...
 8.2.15: Find the sum of each pair of complex numbers. In Exercises 1316, gr...
 8.2.16: Find the sum of each pair of complex numbers. In Exercises 1316, gr...
 8.2.17: Find the sum of each pair of complex numbers. In Exercises 1316, gr...
 8.2.18: Find the sum of each pair of complex numbers. In Exercises 1316, gr...
 8.2.19: Find the sum of each pair of complex numbers. In Exercises 1316, gr...
 8.2.20: Find the sum of each pair of complex numbers. In Exercises 1316, gr...
 8.2.21: Find the sum of each pair of complex numbers. In Exercises 1316, gr...
 8.2.22: Find the sum of each pair of complex numbers. In Exercises 1316, gr...
 8.2.23: Find the sum of each pair of complex numbers. In Exercises 1316, gr...
 8.2.24: Find the sum of each pair of complex numbers. In Exercises 1316, gr...
 8.2.25: Write each complex number in rectangular form. See Example 2. 21cos...
 8.2.26: Write each complex number in rectangular form. See Example 2. 41cos...
 8.2.27: Write each complex number in rectangular form. See Example 2. 101co...
 8.2.28: Write each complex number in rectangular form. See Example 2. 81cos...
 8.2.29: Write each complex number in rectangular form. See Example 2. 41cos...
 8.2.30: Write each complex number in rectangular form. See Example 2. 21cos...
 8.2.31: Write each complex number in rectangular form. See Example 2. 3 cis...
 8.2.32: Write each complex number in rectangular form. See Example 2. 2 cis 30
 8.2.33: Write each complex number in rectangular form. See Example 2. 5 cis...
 8.2.34: Write each complex number in rectangular form. See Example 2. 6 cis...
 8.2.35: Write each complex number in rectangular form. See Example 2. 22 ci...
 8.2.36: Write each complex number in rectangular form. See Example 2. 23 ci...
 8.2.37: Write each complex number in rectangular form. See Example 2. 41cos...
 8.2.38: Write each complex number in rectangular form. See Example 2. 221co...
 8.2.39: Write each complex number in trigonometric form r1cos u + i sin u2,...
 8.2.40: Write each complex number in trigonometric form r1cos u + i sin u2,...
 8.2.41: Write each complex number in trigonometric form r1cos u + i sin u2,...
 8.2.42: Write each complex number in trigonometric form r1cos u + i sin u2,...
 8.2.43: Write each complex number in trigonometric form r1cos u + i sin u2,...
 8.2.44: Write each complex number in trigonometric form r1cos u + i sin u2,...
 8.2.45: Write each complex number in trigonometric form r1cos u + i sin u2,...
 8.2.46: Write each complex number in trigonometric form r1cos u + i sin u2,...
 8.2.47: Write each complex number in trigonometric form r1cos u + i sin u2,...
 8.2.48: Write each complex number in trigonometric form r1cos u + i sin u2,...
 8.2.49: Write each complex number in trigonometric form r1cos u + i sin u2,...
 8.2.50: Write each complex number in trigonometric form r1cos u + i sin u2,...
 8.2.51: Perform each conversion, using a calculator to approximate answers ...
 8.2.52: Perform each conversion, using a calculator to approximate answers ...
 8.2.53: Perform each conversion, using a calculator to approximate answers ...
 8.2.54: Perform each conversion, using a calculator to approximate answers ...
 8.2.55: Perform each conversion, using a calculator to approximate answers ...
 8.2.56: Perform each conversion, using a calculator to approximate answers ...
 8.2.57: Perform each conversion, using a calculator to approximate answers ...
 8.2.58: Perform each conversion, using a calculator to approximate answers ...
 8.2.59: Concept Check The complex number z, where z = x + yi, can be graphe...
 8.2.60: Concept Check The complex number z, where z = x + yi, can be graphe...
 8.2.61: Concept Check The complex number z, where z = x + yi, can be graphe...
 8.2.62: Concept Check The complex number z, where z = x + yi, can be graphe...
 8.2.63: Is z = 0.2i in the Julia set?
 8.2.64: The graph of the Julia set in Figure 11 appears to be symmetric wit...
 8.2.65: Use vectors to show that the conjugate of z is r 3cos1360  u2 + i ...
 8.2.66: Use vectors to show that z = r3cos1u + p2 + i sin1u + p24.
 8.2.67: The difference between two nonreal complex numbers a + bi and c + d...
 8.2.68: The absolute value of the sum of two complex numbers a + bi and c +...
 8.2.69: The absolute value of the difference of two complex numbers a + bi ...
 8.2.70: Show that z and iz have the same absolute value. How are the graphs...
Solutions for Chapter 8.2: Trigonometric (Polar) Form of Complex Numbers
Full solutions for Trigonometry  10th Edition
ISBN: 9780321671776
Solutions for Chapter 8.2: Trigonometric (Polar) Form of Complex Numbers
Get Full SolutionsChapter 8.2: Trigonometric (Polar) Form of Complex Numbers includes 70 full stepbystep solutions. This textbook survival guide was created for the textbook: Trigonometry, edition: 10. Since 70 problems in chapter 8.2: Trigonometric (Polar) Form of Complex Numbers have been answered, more than 34241 students have viewed full stepbystep solutions from this chapter. Trigonometry was written by and is associated to the ISBN: 9780321671776. 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.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

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.

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.

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

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.

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

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.

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

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

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.