 8.5.1: The polar coordinate system consists of a point, called the _____, ...
 8.5.Problem 1: Write (4, 43) in polar coordinates.
 8.5.a: How do you plot the point whose polar coordinates are (3, 45)?
 8.5.2: In polar coordinates, r is the ________ ________ on the terminal si...
 8.5.Problem 2: Graph th
 8.5.b: Why do points in the coordinate plane have more than one representa...
 8.5.3: An angle is considered positive if its terminal side has been rotat...
 8.5.Problem 3: Give
 8.5.c: If you convert (4, 30) to rectangular coordinates, how do you find ...
 8.5.4: To graph a point (r, ) with negative r, plot a point ___ units alon...
 8.5.Problem 4: Convert to rectangular coordinates. a. (2, 60) b. 4, c. 1,
 8.5.d: What are the rectangular coordinates of the point (3, 270)?
 8.5.5: When converting between rectangular and polar coordinates, we assum...
 8.5.Problem 5: Convert to polar coordinates. a. (3, 3) b. (0, 4) c. (3, 1)
 8.5.6: To convert an ordered pair from polar to rectangular coordinates, u...
 8.5.Problem 6: Change r 2 4 cos 2to rectangular coordinates. E
 8.5.7: In rectangular coordinates, each point is represented by a unique o...
 8.5.Problem 7:
 8.5.8: In polar coordinates, each point is represented by a unique ordered...
 8.5.9: (2, 45)
 8.5.10: (3, 60)
 8.5.11: (3, 150)
 8.5.12: (4, 135)
 8.5.13: (1, 225)
 8.5.14: (2, 240)
 8.5.15: (3, 45)
 8.5.16: (4, 60)
 8.5.17: 4,
 8.5.18: 5,
 8.5.19: (2, 0)
 8.5.20: 2,
 8.5.21: (2, 60)
 8.5.22: (1, 30)
 8.5.23: (5, 135)
 8.5.24: (3, 120)
 8.5.25: (3, 30)
 8.5.26: (2, 45)
 8.5.27: (2, 60)
 8.5.28: (2, 60)
 8.5.29: 3,
 8.5.30: (1, )
 8.5.31: (2, 135) 3
 8.5.32: (2, 225) 3
 8.5.33: 43, 34
 8.5.34: 43, U
 8.5.35: (2, 19)
 8.5.36: (3, 124)
 8.5.37: (3, 293)
 8.5.38: (4, 261)
 8.5.39: (3, 3)
 8.5.40: (3, 3)
 8.5.41: (2, 23) 4
 8.5.42: (23, 2)
 8.5.43: (2, 0)
 8.5.44: (2, 0)
 8.5.45: (3, 1) 4
 8.5.46: (1, 3)
 8.5.47: (3, 4)
 8.5.48: (4, 3)
 8.5.49: (1, 2)
 8.5.50: (1, 2)
 8.5.51: (2, 3)
 8.5.52: (3, 2)
 8.5.53: (5, 8)
 8.5.54: (2, 9)
 8.5.55: (1, 6)
 8.5.56: (7, 3)
 8.5.57: r2 9
 8.5.58: r2 4
 8.5.59: r 6 cos 60
 8.5.60: r 6 sin
 8.5.61: r2 4 cos 262
 8.5.62: r2 4 sin 2
 8.5.63: r(cos sin ) 3 64.
 8.5.64: r(cos sin ) 2 W
 8.5.65: x y 5 6
 8.5.66: x y 5
 8.5.67: x2 y2 4 6
 8.5.68: x2 y2 9
 8.5.69: x2 y2 6x 7
 8.5.70: x2 y2 4x
 8.5.71: y x
 8.5.72: y x
 8.5.73: y 6 sin x
 8.5.74: y 6 cos x
 8.5.75: y 4 sin 2x
 8.5.76: y 2 sin 4x
 8.5.77: y 4 2 sin x 78
 8.5.78: y 4 2 cos x
 8.5.79: Which of the following ordered pairs is not a valid representation ...
 8.5.80: Convert 2, to rectangular coordinates. a. (1, 3) b. (3, 1) c. (1, 3...
 8.5.81: Write r 6(cos sin ) in rectangular coordinates. a. x2 y2 6(x y) b. ...
 8.5.82: Write x2 xy y2 1 in polar coordinates, and isolate r if possible. a...
Solutions for Chapter 8.5: Polar Coordinates
Full solutions for Trigonometry  7th Edition
ISBN: 9781111826857
Solutions for Chapter 8.5: Polar Coordinates
Get Full SolutionsSince 93 problems in chapter 8.5: Polar Coordinates have been answered, more than 26062 students have viewed full stepbystep 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 stepbystep solutions.

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 = 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.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

Incidence matrix of a directed graph.
The m by n edgenode 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 eiO , 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.