 6.3.1: Exer. 14: A point P(x, y) is shown on the unit circle U correspondi...
 6.3.2: Exer. 14: A point P(x, y) is shown on the unit circle U correspondi...
 6.3.3: Exer. 14: A point P(x, y) is shown on the unit circle U correspondi...
 6.3.4: Exer. 14: A point P(x, y) is shown on the unit circle U correspondi...
 6.3.5: Exer. 58: Let P(t) be the point on the unit circle U that correspon...
 6.3.6: Exer. 58: Let P(t) be the point on the unit circle U that correspon...
 6.3.7: Exer. 58: Let P(t) be the point on the unit circle U that correspon...
 6.3.8: Exer. 58: Let P(t) be the point on the unit circle U that correspon...
 6.3.9: Exer. 916: Let P be the point on the unit circle U that corresponds...
 6.3.10: Exer. 916: Let P be the point on the unit circle U that corresponds...
 6.3.11: Exer. 916: Let P be the point on the unit circle U that corresponds...
 6.3.12: Exer. 916: Let P be the point on the unit circle U that corresponds...
 6.3.13: Exer. 916: Let P be the point on the unit circle U that corresponds...
 6.3.14: Exer. 916: Let P be the point on the unit circle U that corresponds...
 6.3.15: Exer. 916: Let P be the point on the unit circle U that corresponds...
 6.3.16: Exer. 916: Let P be the point on the unit circle U that corresponds...
 6.3.17: Exer. 1720: Use a formula for negatives to find the exact value.
 6.3.18: Exer. 1720: Use a formula for negatives to find the exact value.
 6.3.19: Exer. 1720: Use a formula for negatives to find the exact value.
 6.3.20: Exer. 1720: Use a formula for negatives to find the exact value.
 6.3.21: Exer. 2126: Verify the identity by transforming the lefthand side i...
 6.3.22: Exer. 2126: Verify the identity by transforming the lefthand side i...
 6.3.23: Exer. 2126: Verify the identity by transforming the lefthand side i...
 6.3.24: Exer. 2126: Verify the identity by transforming the lefthand side i...
 6.3.25: Exer. 2126: Verify the identity by transforming the lefthand side i...
 6.3.26: Exer. 2126: Verify the identity by transforming the lefthand side i...
 6.3.27: Exer. 2738: Complete the statement by referring to a graph of a tri...
 6.3.28: Exer. 2738: Complete the statement by referring to a graph of a tri...
 6.3.29: Exer. 2738: Complete the statement by referring to a graph of a tri...
 6.3.30: Exer. 2738: Complete the statement by referring to a graph of a tri...
 6.3.31: Exer. 2738: Complete the statement by referring to a graph of a tri...
 6.3.32: Exer. 2738: Complete the statement by referring to a graph of a tri...
 6.3.33: Exer. 2738: Complete the statement by referring to a graph of a tri...
 6.3.34: Exer. 2738: Complete the statement by referring to a graph of a tri...
 6.3.35: Exer. 2738: Complete the statement by referring to a graph of a tri...
 6.3.36: Exer. 2738: Complete the statement by referring to a graph of a tri...
 6.3.37: Exer. 2738: Complete the statement by referring to a graph of a tri...
 6.3.38: Exer. 2738: Complete the statement by referring to a graph of a tri...
 6.3.39: Exer. 3946: Refer to the graph of y sin x or y cos x to find the ex...
 6.3.40: Exer. 3946: Refer to the graph of y sin x or y cos x to find the ex...
 6.3.41: Exer. 3946: Refer to the graph of y sin x or y cos x to find the ex...
 6.3.42: Exer. 3946: Refer to the graph of y sin x or y cos x to find the ex...
 6.3.43: Exer. 3946: Refer to the graph of y sin x or y cos x to find the ex...
 6.3.44: Exer. 3946: Refer to the graph of y sin x or y cos x to find the ex...
 6.3.45: Exer. 3946: Refer to the graph of y sin x or y cos x to find the ex...
 6.3.46: Exer. 3946: Refer to the graph of y sin x or y cos x to find the ex...
 6.3.47: Exer. 4750: Refer to the graph of y tan x to find the exact values ...
 6.3.48: Exer. 4750: Refer to the graph of y tan x to find the exact values ...
 6.3.49: Exer. 4750: Refer to the graph of y tan x to find the exact values ...
 6.3.50: Exer. 4750: Refer to the graph of y tan x to find the exact values ...
 6.3.51: Exer. 5154: Refer to the graph of the equation on the specified int...
 6.3.52: Exer. 5154: Refer to the graph of the equation on the specified int...
 6.3.53: Exer. 5154: Refer to the graph of the equation on the specified int...
 6.3.54: Exer. 5154: Refer to the graph of the equation on the specified int...
 6.3.55: Exer. 5562: Use the graph of a trigonometric function to sketch the...
 6.3.56: Exer. 5562: Use the graph of a trigonometric function to sketch the...
 6.3.57: Exer. 5562: Use the graph of a trigonometric function to sketch the...
 6.3.58: Exer. 5562: Use the graph of a trigonometric function to sketch the...
 6.3.59: Exer. 5562: Use the graph of a trigonometric function to sketch the...
 6.3.60: Exer. 5562: Use the graph of a trigonometric function to sketch the...
 6.3.61: Exer. 5562: Use the graph of a trigonometric function to sketch the...
 6.3.62: Exer. 5562: Use the graph of a trigonometric function to sketch the...
 6.3.63: Exer. 6366: Find the intervals between 2p and 2p on which the given...
 6.3.64: Exer. 6366: Find the intervals between 2p and 2p on which the given...
 6.3.65: Exer. 6366: Find the intervals between 2p and 2p on which the given...
 6.3.66: Exer. 6366: Find the intervals between 2p and 2p on which the given...
 6.3.67: Practice sketching the graph of the sine function, taking different...
 6.3.68: Work Exercise 67 for the cosecant, secant, and cotangent functions.
 6.3.69: Exer. 6972: Use the figure to approximate the following to one deci...
 6.3.70: Exer. 6972: Use the figure to approximate the following to one deci...
 6.3.71: Exer. 6972: Use the figure to approximate the following to one deci...
 6.3.72: Exer. 6972: Use the figure to approximate the following to one deci...
 6.3.73: On March 17, 1981, in Tucson, Arizona, the temperature in degrees F...
 6.3.74: Trigonometric functions are used extensively in the design of indus...
Solutions for Chapter 6.3: Trigonometric Functions of Real Numbers
Full solutions for Algebra and Trigonometry with Analytic Geometry  12th Edition
ISBN: 9780495559719
Solutions for Chapter 6.3: Trigonometric Functions of Real Numbers
Get Full SolutionsThis textbook survival guide was created for the textbook: Algebra and Trigonometry with Analytic Geometry, edition: 12. Since 74 problems in chapter 6.3: Trigonometric Functions of Real Numbers have been answered, more than 33453 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Algebra and Trigonometry with Analytic Geometry was written by and is associated to the ISBN: 9780495559719. Chapter 6.3: Trigonometric Functions of Real Numbers includes 74 full stepbystep solutions.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

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.)

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.

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

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.

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.

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

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

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 .

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

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.

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

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

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.