 8.5.1: Concept Check For each point given in polar coordinates, state the ...
 8.5.2: Concept Check For each point given in polar coordinates, state the ...
 8.5.3: For each pair of polar coordinates, (a) plot the point, (b) give tw...
 8.5.4: For each pair of polar coordinates, (a) plot the point, (b) give tw...
 8.5.5: For each pair of polar coordinates, (a) plot the point, (b) give tw...
 8.5.6: For each pair of polar coordinates, (a) plot the point, (b) give tw...
 8.5.7: For each pair of polar coordinates, (a) plot the point, (b) give tw...
 8.5.8: For each pair of polar coordinates, (a) plot the point, (b) give tw...
 8.5.9: For each pair of polar coordinates, (a) plot the point, (b) give tw...
 8.5.10: For each pair of polar coordinates, (a) plot the point, (b) give tw...
 8.5.11: For each pair of polar coordinates, (a) plot the point, (b) give tw...
 8.5.12: For each pair of polar coordinates, (a) plot the point, (b) give tw...
 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 rectangular coordinates, (a) plot the point and (b...
 8.5.16: For each pair of rectangular coordinates, (a) plot the point and (b...
 8.5.17: For each pair of rectangular coordinates, (a) plot the point and (b...
 8.5.18: For each pair of rectangular coordinates, (a) plot the point and (b...
 8.5.19: For each pair of rectangular coordinates, (a) plot the point and (b...
 8.5.20: For each pair of rectangular coordinates, (a) plot the point and (b...
 8.5.21: For each pair of rectangular coordinates, (a) plot the point and (b...
 8.5.22: For each pair of rectangular coordinates, (a) plot the point and (b...
 8.5.23: For each pair of rectangular coordinates, (a) plot the point and (b...
 8.5.24: For each pair of rectangular coordinates, (a) plot the point and (b...
 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 rectangular equation, give its equivalent polar equation a...
 8.5.28: For each rectangular equation, give its equivalent polar equation a...
 8.5.29: For each rectangular equation, give its equivalent polar equation a...
 8.5.30: For each rectangular equation, give its equivalent polar equation a...
 8.5.31: For each rectangular equation, give its equivalent polar equation a...
 8.5.32: For each rectangular equation, give its equivalent polar equation a...
 8.5.33: Begin with the equation y = k, whose graph is a horizontal line. Ma...
 8.5.34: Solve the equation in Exercise 33 for r.
 8.5.35: Rewrite the equation in Exercise 34 using the appropriate reciproca...
 8.5.36: Sketch the graph of the equation r = 3 csc u. What is the correspon...
 8.5.37: Begin with the equation x = k, whose graph is a vertical line. Make...
 8.5.38: Solve the equation in Exercise 37 for r.
 8.5.39: Rewrite the equation in Exercise 38 using the appropriate reciproca...
 8.5.40: Sketch the graph of r = 3 sec u. What is the corresponding rectangu...
 8.5.41: Concept Check In Exercises 4144, match each equation with its polar...
 8.5.42: Concept Check In Exercises 4144, match each equation with its polar...
 8.5.43: Concept Check In Exercises 4144, match each equation with its polar...
 8.5.44: Concept Check In Exercises 4144, match each equation with its polar...
 8.5.45: Give a complete graph of each polar equation. In Exercises 4554, al...
 8.5.46: Give a complete graph of each polar equation. In Exercises 4554, al...
 8.5.47: Give a complete graph of each polar equation. In Exercises 4554, al...
 8.5.48: Give a complete graph of each polar equation. In Exercises 4554, al...
 8.5.49: Give a complete graph of each polar equation. In Exercises 4554, al...
 8.5.50: Give a complete graph of each polar equation. In Exercises 4554, al...
 8.5.51: Give a complete graph of each polar equation. In Exercises 4554, al...
 8.5.52: Give a complete graph of each polar equation. In Exercises 4554, al...
 8.5.53: Give a complete graph of each polar equation. In Exercises 4554, al...
 8.5.54: Give a complete graph of each polar equation. In Exercises 4554, al...
 8.5.55: Give a complete graph of each polar equation. In Exercises 4554, al...
 8.5.56: Give a complete graph of each polar equation. In Exercises 4554, al...
 8.5.57: For each equation, find an equivalent equation in rectangular coord...
 8.5.58: For each equation, find an equivalent equation in rectangular coord...
 8.5.59: For each equation, find an equivalent equation in rectangular coord...
 8.5.60: For each equation, find an equivalent equation in rectangular coord...
 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: Graph r = u, a spiral of Archimedes. (See Example 7.) Use both posi...
 8.5.68: Use a graphing calculator window of 31250, 12504 by 31250, 12504,...
 8.5.69: Find the polar equation of the line that passes through the points ...
 8.5.70: Explain how to plot a point 1r, u2 in polar coordinates, if r 6 0.
 8.5.71: Complete the missing ordered pairs in the graphs below. (a) (b) (c)
 8.5.72: Based on your results in Exercise 71, fill in the blanks with the c...
 8.5.73: The graph of r = au in polar coordinates is an example of the spira...
 8.5.74: The graph of r = au in polar coordinates is an example of the spira...
 8.5.75: The graph of r = au in polar coordinates is an example of the spira...
 8.5.76: The graph of r = au in polar coordinates is an example of the spira...
 8.5.77: Find the polar coordinates of the points of intersection of the giv...
 8.5.78: Find the polar coordinates of the points of intersection of the giv...
 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: Orbits of Satellites The polar equation r = a11  e22 1 + e cos u c...
 8.5.82: Radio Towers and Broadcasting Patterns Many times radio stations do...
Solutions for Chapter 8.5: Polar Equations and Graphs
Full solutions for Trigonometry  10th Edition
ISBN: 9780321671776
Solutions for Chapter 8.5: Polar Equations and Graphs
Get Full SolutionsThis textbook survival guide was created for the textbook: Trigonometry, edition: 10. Chapter 8.5: Polar Equations and Graphs includes 82 full stepbystep solutions. Since 82 problems in chapter 8.5: Polar Equations and Graphs have been answered, more than 33220 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.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

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.

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.

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

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.

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

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.

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

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

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

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

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 ().

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

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

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