 8.5.1: Fill in the blank to correctly complete each sentence. For the pola...
 8.5.2: Fill in the blank to correctly complete each sentence. For the pola...
 8.5.3: Fill in the blank to correctly complete each sentence. For the pola...
 8.5.4: Fill in the blank to correctly complete each sentence. For the pola...
 8.5.5: For each point given in polar coordinates, state the quadrant in wh...
 8.5.6: For each point given in polar coordinates, state the quadrant in wh...
 8.5.7: For each point given in polar coordinates, state the quadrant in wh...
 8.5.8: For each point given in polar coordinates, state the quadrant in wh...
 8.5.9: For each point given in polar coordinates, state the axis on which ...
 8.5.10: For each point given in polar coordinates, state the axis on which ...
 8.5.11: For each point given in polar coordinates, state the axis on which ...
 8.5.12: For each point given in polar coordinates, state the axis on which ...
 8.5.13: For each pair of polar coordinates, (a) plot the point, (b) give tw...
 8.5.14: For each pair of polar coordinates, (a) plot the point, (b) give tw...
 8.5.15: For each pair of polar coordinates, (a) plot the point, (b) give tw...
 8.5.16: For each pair of polar coordinates, (a) plot the point, (b) give tw...
 8.5.17: For each pair of polar coordinates, (a) plot the point, (b) give tw...
 8.5.18: For each pair of polar coordinates, (a) plot the point, (b) give tw...
 8.5.19: For each pair of polar coordinates, (a) plot the point, (b) give tw...
 8.5.20: For each pair of polar coordinates, (a) plot the point, (b) give tw...
 8.5.21: For each pair of polar coordinates, (a) plot the point, (b) give tw...
 8.5.22: For each pair of polar coordinates, (a) plot the point, (b) give tw...
 8.5.23: For each pair of polar coordinates, (a) plot the point, (b) give tw...
 8.5.24: For each pair of polar coordinates, (a) plot the point, (b) give tw...
 8.5.25: For each pair of rectangular coordinates, (a) plot the point and (b...
 8.5.26: For each pair of rectangular coordinates, (a) plot the point and (b...
 8.5.27: For each pair of rectangular coordinates, (a) plot the point and (b...
 8.5.28: For each pair of rectangular coordinates, (a) plot the point and (b...
 8.5.29: For each pair of rectangular coordinates, (a) plot the point and (b...
 8.5.30: For each pair of rectangular coordinates, (a) plot the point and (b...
 8.5.31: For each pair of rectangular coordinates, (a) plot the point and (b...
 8.5.32: For each pair of rectangular coordinates, (a) plot the point and (b...
 8.5.33: For each pair of rectangular coordinates, (a) plot the point and (b...
 8.5.34: For each pair of rectangular coordinates, (a) plot the point and (b...
 8.5.35: For each pair of rectangular coordinates, (a) plot the point and (b...
 8.5.36: For each pair of rectangular coordinates, (a) plot the point and (b...
 8.5.37: For each rectangular equation, give the equivalent polar equation a...
 8.5.38: For each rectangular equation, give the equivalent polar equation a...
 8.5.39: For each rectangular equation, give the equivalent polar equation a...
 8.5.40: For each rectangular equation, give the equivalent polar equation a...
 8.5.41: For each rectangular equation, give the equivalent polar equation a...
 8.5.42: For each rectangular equation, give the equivalent polar equation a...
 8.5.43: Match each equation with its polar graph from choices AD.r = 3
 8.5.44: Match each equation with its polar graph from choices AD. r = cos 3u
 8.5.45: Match each equation with its polar graph from choices AD.r = cos 2u
 8.5.46: Match each equation with its polar graph from choices AD.r =2 cos u...
 8.5.47: Graph each polar equation. In Exercises 4756, also identify the typ...
 8.5.48: Graph each polar equation. In Exercises 4756, also identify the typ...
 8.5.49: Graph each polar equation. In Exercises 4756, also identify the typ...
 8.5.50: Graph each polar equation. In Exercises 4756, also identify the typ...
 8.5.51: Graph each polar equation. In Exercises 4756, also identify the typ...
 8.5.52: Graph each polar equation. In Exercises 4756, also identify the typ...
 8.5.53: Graph each polar equation. In Exercises 4756, also identify the typ...
 8.5.54: Graph each polar equation. In Exercises 4756, also identify the typ...
 8.5.55: Graph each polar equation. In Exercises 4756, also identify the typ...
 8.5.56: Graph each polar equation. In Exercises 4756, also identify the typ...
 8.5.57: r = 2 sin u tan u (This is a cissoid.)
 8.5.58: r =cos 2u cos u (This is a cissoid with a loop.)
 8.5.59: Graph each spiral of Archimedes. See Example 7. r = u (Use both pos...
 8.5.60: Graph each spiral of Archimedes. See Example 7. 4u (Use a graphing ...
 8.5.61: For each equation, find an equivalent equation in rectangular coord...
 8.5.62: For each equation, find an equivalent equation in rectangular coord...
 8.5.63: For each equation, find an equivalent equation in rectangular coord...
 8.5.64: For each equation, find an equivalent equation in rectangular coord...
 8.5.65: For each equation, find an equivalent equation in rectangular coord...
 8.5.66: For each equation, find an equivalent equation in rectangular coord...
 8.5.67: For each equation, find an equivalent equation in rectangular coord...
 8.5.68: For each equation, find an equivalent equation in rectangular coord...
 8.5.69: For each equation, find an equivalent equation in rectangular coord...
 8.5.70: For each equation, find an equivalent equation in rectangular coord...
 8.5.71: Solve each problem. Find the polar equation of the line that passes...
 8.5.72: Solve each problem.12, 902. Explain how to plot a point 1r, u2 in p...
 8.5.73: The polar graphs in this section exhibit symmetry. Visualize an xy...
 8.5.74: The polar graphs in this section exhibit symmetry. Visualize an xy...
 8.5.75: The graph of r = au in polar coordinates is an example of a spiral ...
 8.5.76: The graph of r = au in polar coordinates is an example of a spiral ...
 8.5.77: The graph of r = au in polar coordinates is an example of a spiral ...
 8.5.78: The graph of r = au in polar coordinates is an example of a spiral ...
 8.5.79: Find the polar coordinates of the points of intersection of the giv...
 8.5.80: Find the polar coordinates of the points of intersection of the giv...
 8.5.81: Find the polar coordinates of the points of intersection of the giv...
 8.5.82: Find the polar coordinates of the points of intersection of the giv...
 8.5.83: Orbits of Satellites The polar equationr =a11  e 22 1 + e cos u ca...
 8.5.84: Radio Towers and Broadcasting Patterns Radio stations do not always...
 8.5.85: In rectangular coordinates, the graph of ax + by = c is a horizonta...
 8.5.86: In rectangular coordinates, the graph of ax + by = c is a horizonta...
 8.5.87: In rectangular coordinates, the graph of ax + by = c is a horizonta...
 8.5.88: In rectangular coordinates, the graph of ax + by = c is a horizonta...
 8.5.89: In rectangular coordinates, the graph of ax + by = c is a horizonta...
 8.5.90: In rectangular coordinates, the graph of ax + by = c is a horizonta...
 8.5.91: In rectangular coordinates, the graph of ax + by = c is a horizonta...
 8.5.92: In rectangular coordinates, the graph of ax + by = c is a horizonta...
Solutions for Chapter 8.5: Polar Equations and Graphs
Full solutions for Trigonometry  11th Edition
ISBN: 9780134217437
Solutions for Chapter 8.5: Polar Equations and Graphs
Get Full SolutionsChapter 8.5: Polar Equations and Graphs includes 92 full stepbystep solutions. Since 92 problems in chapter 8.5: Polar Equations and Graphs have been answered, more than 19480 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Trigonometry, edition: 11. Trigonometry was written by and is associated to the ISBN: 9780134217437. This expansive textbook survival guide covers the following chapters and their 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.

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

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.

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.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

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

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.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

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.

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.

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

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.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

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.

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

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