 8.1: Write each number as the product of a real number and i. 29
 8.2: Write each number as the product of a real number and i. 212
 8.3: Solve each quadratic equation over the set of complex numbers. x2 =...
 8.4: Solve each quadratic equation over the set of complex numbers. x12x...
 8.5: Perform each operation. Write answers in rectangular form. 11  i2 ...
 8.6: Perform each operation. Write answers in rectangular form. 12  5i2...
 8.7: Perform each operation. Write answers in rectangular form. 16  5i2...
 8.8: Perform each operation. Write answers in rectangular form. 14  2i2...
 8.9: Perform each operation. Write answers in rectangular form. 13 + 5i2...
 8.10: Perform each operation. Write answers in rectangular form. 14  i21...
 8.11: Perform each operation. Write answers in rectangular form. 12 + 6i22
 8.12: Perform each operation. Write answers in rectangular form. 16  3i22
 8.13: Perform each operation. Write answers in rectangular form. 11  i23
 8.14: Perform each operation. Write answers in rectangular form. 12 + i23
 8.15: Perform each operation. Write answers in rectangular form. 25  19i...
 8.16: Perform each operation. Write answers in rectangular form. 2  5i 1...
 8.17: Perform each operation. Write answers in rectangular form. 2 + i 1 ...
 8.18: Perform each operation. Write answers in rectangular form. 3 + 2i i
 8.19: Perform each operation. Write answers in rectangular form. i 53
 8.20: Perform each operation. Write answers in rectangular form. i 41
 8.21: Perform each operation. Write answers in rectangular form. 351cos 9...
 8.22: Perform each operation. Write answers in rectangular form. 33 cis 1...
 8.23: Perform each operation. Write answers in rectangular form. 21cos 60...
 8.24: Perform each operation. Write answers in rectangular form. 4 cis 27...
 8.25: Perform each operation. Write answers in rectangular form. A23 + i B3
 8.26: Perform each operation. Write answers in rectangular form. 12  2i25
 8.27: Perform each operation. Write answers in rectangular form. 1cos 100...
 8.28: Concept Check The vector representing a real number will lie on the...
 8.29: Graph each complex number as a vector. 5i
 8.30: Graph each complex number as a vector. 4 + 2i
 8.31: Graph each complex number as a vector. 3  3i23
 8.32: Find the sum of 7 + 3i and 2 + i. Graph both complex numbers and t...
 8.33: Perform each conversion, using a calculator to approximate answers ...
 8.34: Perform each conversion, using a calculator to approximate answers ...
 8.35: Perform each conversion, using a calculator to approximate answers ...
 8.36: Perform each conversion, using a calculator to approximate answers ...
 8.37: Perform each conversion, using a calculator to approximate answers ...
 8.38: Perform each conversion, using a calculator to approximate answers ...
 8.39: Perform each conversion, using a calculator to approximate answers ...
 8.40: Perform each conversion, using a calculator to approximate answers ...
 8.41: Concept Check The complex number z, where z = x + yi, can be graphe...
 8.42: Concept Check The complex number z, where z = x + yi, can be graphe...
 8.43: Concept Check The complex number z, where z = x + yi, can be graphe...
 8.44: Concept Check The complex number z, where z = x + yi, can be graphe...
 8.45: Concept Check The complex number z, where z = x + yi, can be graphe...
 8.46: Concept Check The complex number z, where z = x + yi, can be graphe...
 8.47: Solve each equation. Leave answers in trigonometric form. x4 + 16 = 0
 8.48: Solve each equation. Leave answers in trigonometric form. x3 + 125 = 0
 8.49: Solve each equation. Leave answers in trigonometric form. x2 + i = 0
 8.50: Convert 15, 3152 to rectangular coordinates.
 8.51: Convert A1, 23 B to polar coordinates, with 0 u 6 360 and r 7 0.
 8.52: Concept Check What will the graph of r = k be, for k 7 0?
 8.53: Identify and graph each polar equation for u in 30, 3602. r = 4 cos u
 8.54: Identify and graph each polar equation for u in 30, 3602. r = 1 + ...
 8.55: Identify and graph each polar equation for u in 30, 3602. r = 2 sin 4u
 8.56: Identify and graph each polar equation for u in 30, 3602. r = 2 2 c...
 8.57: Find an equivalent equation in rectangular coordinates. r = 3 1 + c...
 8.58: Find an equivalent equation in rectangular coordinates. r = sin u +...
 8.59: Find an equivalent equation in rectangular coordinates. r = 2
 8.60: Find an equivalent equation in polar coordinates. y = x
 8.61: Find an equivalent equation in polar coordinates. y = x2
 8.62: Find an equivalent equation in polar coordinates. x2 + y2 = 25
 8.63: In Exercises 6366, identify the geometric symmetry (A, B, or C) tha...
 8.64: In Exercises 6366, identify the geometric symmetry (A, B, or C) tha...
 8.65: In Exercises 6366, identify the geometric symmetry (A, B, or C) tha...
 8.66: In Exercises 6366, identify the geometric symmetry (A, B, or C) tha...
 8.67: In Exercises 6770, find a polar equation having the given graph.
 8.68: In Exercises 6770, find a polar equation having the given graph.
 8.69: In Exercises 6770, find a polar equation having the given graph.
 8.70: In Exercises 6770, find a polar equation having the given graph.
 8.71: Graph the plane curve defined by the parametric equations x = t + c...
 8.72: Show that the distance between 1r1, u12 and 1r2, u22 in polar coord...
 8.73: Find a rectangular equation for each plane curve with the given par...
 8.74: Find a rectangular equation for each plane curve with the given par...
 8.75: Find a rectangular equation for each plane curve with the given par...
 8.76: Find a rectangular equation for each plane curve with the given par...
 8.77: Find a rectangular equation for each plane curve with the given par...
 8.78: Find a pair of parametric equations whose graph is the circle havin...
 8.79: Flight of a Baseball A batter hits a baseball when it is 3.2 ft abo...
 8.80: Mandelbrot Set Consider the complex number z = 1 + i. Compute the v...
Solutions for Chapter 8: Review Exercises
Full solutions for Trigonometry  10th Edition
ISBN: 9780321671776
Solutions for Chapter 8: Review Exercises
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Trigonometry, edition: 10. Chapter 8: Review Exercises includes 80 full stepbystep solutions. Since 80 problems in chapter 8: Review Exercises have been answered, more than 35614 students have viewed full stepbystep solutions from this chapter. Trigonometry was written by and is associated to the ISBN: 9780321671776.

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

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

Column space C (A) =
space of all combinations of the columns of A.

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

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

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

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

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.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.