 8.6.Problem 1: Sketch the graph of r 4 cos .
 8.6.a: What is one way to graph an equation written in polar coordinates?
 8.6.1: The simplest method of graphing a polar equation is to plot points ...
 8.6.2: Another way to graph a polar equation is to convert it to _________...
 8.6.Problem 2: Sketch the graph of r 2 cos 2.
 8.6.b: What type of geometric figure is the graph of r 6 sin ?
 8.6.3: A more advanced method of graphing a polar equation is to first gra...
 8.6.Problem 3: Sketch
 8.6.c: What is the largest value of r that can be obtained from the equati...
 8.6.4: Match each equation with its appropriate graph. Assume a and b are ...
 8.6.d: What type of curve is the graph of r 4 2 sin ? 8.
 8.6.5: r 6 cos 6.
 8.6.6: r 4 sin
 8.6.7: r sin 28.
 8.6.8: r cos 2
 8.6.9: r 2 sin 210
 8.6.10: r 2 cos 2
 8.6.11: 3 3 sin 12.
 8.6.12: r 3 3 cos D
 8.6.13: r 5 sin 314
 8.6.14: 2 6 sin 2
 8.6.15: r 4 4 cos 16.
 8.6.16: r 6
 8.6.17: r 4
 8.6.18: r 5 3 cos 1
 8.6.19: r 2 3 cos 220
 8.6.20: r 3 cos 2
 8.6.21:
 8.6.22: r 5 5 sin 2
 8.6.23: r 3 5 sin 24.
 8.6.24:
 8.6.25: r 3 2
 8.6.26: r 2
 8.6.27:
 8.6.28:
 8.6.29: r 3 sin 30
 8.6.30: r 3 cos
 8.6.31: r 4 2 sin 32.
 8.6.32: r 4 2 cos 3
 8.6.33: r 2 4 cos 34.
 8.6.34: r 2 4 sin 3
 8.6.35: r 2 2 sin 36.
 8.6.36: r 2 2 cos 3
 8.6.37: r2 4 cos 238
 8.6.38: r2 9 sin 2
 8.6.39: r 2 sin 240
 8.6.40: r 2 cos 2
 8.6.41: r 4 cos 342
 8.6.42: r 4 sin 3
 8.6.43: r 2 cos 244
 8.6.44: r 2 sin 2
 8.6.45: r 4 sin 546
 8.6.46: r 6 cos 6
 8.6.47: r 3 3 cos 48.
 8.6.48: r 3 3 sin 4
 8.6.49: r 1 4 cos 50
 8.6.50: r 4 5 sin
 8.6.51: r 2 cos 23 sin 52.
 8.6.52: r 3 sin 22 cos 53
 8.6.53: r 3 sin 2sin 54.
 8.6.54: r 2 cos 2cos C
 8.6.55: x2 y2 16 5
 8.6.56: x2 y2 25
 8.6.57: x2 y2 6x 5
 8.6.58: x2 y2 6y
 8.6.59:
 8.6.60:
 8.6.61: r(2 cos 3 sin ) 6 62.
 8.6.62: r(3 cos 2 sin ) 6 6
 8.6.63: r(1 cos ) 1 6
 8.6.64: r(1 sin ) 1
 8.6.65: r 4 sin 66
 8.6.66: r 6 cos
 8.6.67: Graph r1 2 sin and r2 2 cos and then name two points they have in c...
 8.6.68: Graph r1 2 2 cos and r2 2 2 cos and name three points they have in ...
 8.6.69: y sin x cos x, 0 x 470
 8.6.70: y cos x sin x, 0 x 4
 8.6.71: y x sin x, 0 x 8 72.
 8.6.72: y x cos x, 0 x 8 7
 8.6.73: y 3 sin x cos 2x, 0 x 474.
 8.6.74: y sin x cos 2x, 0 x 4
 8.6.75: Table 4 shows ordered pairs for a polar equation. Use the data in t...
 8.6.76: Use the rectangular graph of r f() shown in Figure 22 to sketch the...
 8.6.77: Use your graphing calculator to determine which of the following eq...
 8.6.78: Which equation has a graph that is a fourleaved rose? a. r 3 cos 4...
Solutions for Chapter 8.6: Equations in Polar Coordinates and Their Graphs
Full solutions for Trigonometry  7th Edition
ISBN: 9781111826857
Solutions for Chapter 8.6: Equations in Polar Coordinates and Their Graphs
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Trigonometry was written by and is associated to the ISBN: 9781111826857. This textbook survival guide was created for the textbook: Trigonometry, edition: 7. Since 85 problems in chapter 8.6: Equations in Polar Coordinates and Their Graphs have been answered, more than 24949 students have viewed full stepbystep solutions from this chapter. Chapter 8.6: Equations in Polar Coordinates and Their Graphs includes 85 full stepbystep solutions.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

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

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

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.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

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 .

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.

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.

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Outer product uv T
= column times row = rank one matrix.

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

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

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.