×
Log in to StudySoup
Get Full Access to Trigonometry - Textbook Survival Guide
Join StudySoup for FREE
Get Full Access to Trigonometry - Textbook Survival Guide

Solutions for Chapter 8.5: Polar Coordinates

Trigonometry | 7th Edition | ISBN: 9781111826857 | Authors: Charles P. McKeague

Full solutions for Trigonometry | 7th Edition

ISBN: 9781111826857

Trigonometry | 7th Edition | ISBN: 9781111826857 | Authors: Charles P. McKeague

Solutions for Chapter 8.5: Polar Coordinates

Solutions for Chapter 8.5
4 5 0 336 Reviews
27
1
Textbook: Trigonometry
Edition: 7
Author: Charles P. McKeague
ISBN: 9781111826857

Since 93 problems in chapter 8.5: Polar Coordinates have been answered, more than 26062 students have viewed full step-by-step solutions from this chapter. Trigonometry was written by and is associated to the ISBN: 9781111826857. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Trigonometry, edition: 7. Chapter 8.5: Polar Coordinates includes 93 full step-by-step solutions.

Key Math Terms and definitions covered in this textbook
  • Associative Law (AB)C = A(BC).

    Parentheses can be removed to leave ABC.

  • Block matrix.

    A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

  • Cyclic shift

    S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

  • Diagonalization

    A = S-1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k S-I.

  • Hessenberg matrix H.

    Triangular matrix with one extra nonzero adjacent diagonal.

  • Incidence matrix of a directed graph.

    The m by n edge-node incidence matrix has a row for each edge (node i to node j), with entries -1 and 1 in columns i and j .

  • Left nullspace N (AT).

    Nullspace of AT = "left nullspace" of A because y T A = OT.

  • Length II x II.

    Square root of x T x (Pythagoras in n dimensions).

  • 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.

  • Matrix multiplication AB.

    The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

  • Minimal polynomial of A.

    The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A - AI) if no eigenvalues are repeated; always meA) divides peA).

  • Partial pivoting.

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

  • Pivot columns of A.

    Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

  • Right inverse A+.

    If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.

  • Rotation matrix

    R = [~ CS ] rotates the plane by () and R- 1 = RT rotates back by -(). Eigenvalues are eiO and e-iO , eigenvectors are (1, ±i). c, s = cos (), sin ().

  • Special solutions to As = O.

    One free variable is Si = 1, other free variables = o.

  • 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·

  • Vector addition.

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

  • Volume of box.

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

×
Log in to StudySoup
Get Full Access to Trigonometry - Textbook Survival Guide
Join StudySoup for FREE
Get Full Access to Trigonometry - Textbook Survival Guide
×
Reset your password