 5.1: For each expression in Column I, choose the expression from Column ...
 5.2: For each expression in Column I, choose the expression from Column ...
 5.3: For each expression in Column I, choose the expression from Column ...
 5.4: For each expression in Column I, choose the expression from Column ...
 5.5: For each expression in Column I, choose the expression from Column ...
 5.6: For each expression in Column I, choose the expression from Column ...
 5.7: Use identities to write each expression in terms of sin u and cos u...
 5.8: Use identities to write each expression in terms of sin u and cos u...
 5.9: Use identities to write each expression in terms of sin u and cos u...
 5.10: Use identities to write each expression in terms of sin u and cos u...
 5.11: Use identities to write each expression in terms of sin u and cos u...
 5.12: Use identities to write each expression in terms of sin u and cos u...
 5.13: Work each problem. Use the trigonometric identities to fnd sin x, t...
 5.14: Work each problem. Given tan x =  5 4 , where p 2 6 x 6 p, use the...
 5.15: Work each problem. Find the exact values of the six trigonometric f...
 5.16: Work each problem. Find the exact values of sin x, cos x, and tan x...
 5.17: For each expression in Column I, use an identity to choose an expre...
 5.18: For each expression in Column I, use an identity to choose an expre...
 5.19: For each expression in Column I, use an identity to choose an expre...
 5.20: For each expression in Column I, use an identity to choose an expre...
 5.21: For each expression in Column I, use an identity to choose an expre...
 5.22: For each expression in Column I, use an identity to choose an expre...
 5.23: For each expression in Column I, use an identity to choose an expre...
 5.24: For each expression in Column I, use an identity to choose an expre...
 5.25: For each expression in Column I, use an identity to choose an expre...
 5.26: For each expression in Column I, use an identity to choose an expre...
 5.27: Use the given information to fnd sin1x + y2, cos1x  y2, tan1x + y2...
 5.28: Use the given information to fnd sin1x + y2, cos1x  y2, tan1x + y2...
 5.29: Use the given information to fnd sin1x + y2, cos1x  y2, tan1x + y2...
 5.30: Use the given information to fnd sin1x + y2, cos1x  y2, tan1x + y2...
 5.31: Use the given information to fnd sin1x + y2, cos1x  y2, tan1x + y2...
 5.32: Use the given information to fnd sin1x + y2, cos1x  y2, tan1x + y2...
 5.33: Find values of the sine and cosine functions for each angle measure...
 5.34: Find values of the sine and cosine functions for each angle measure...
 5.35: Find values of the sine and cosine functions for each angle measure...
 5.36: Find values of the sine and cosine functions for each angle measure...
 5.37: Use the given information to fnd each of the following. cos u 2 , g...
 5.38: Use the given information to fnd each of the following. sin A 2 , g...
 5.39: Use the given information to fnd each of the following. tan x, give...
 5.40: Use the given information to fnd each of the following. sin y, give...
 5.41: Use the given information to fnd each of the following. tan x 2 , g...
 5.42: Use the given information to fnd each of the following. sin 2x, giv...
 5.43: Graph each expression and use the graph to make a conjecture, predi...
 5.44: Graph each expression and use the graph to make a conjecture, predi...
 5.45: Graph each expression and use the graph to make a conjecture, predi...
 5.46: Graph each expression and use the graph to make a conjecture, predi...
 5.47: Graph each expression and use the graph to make a conjecture, predi...
 5.48: Graph each expression and use the graph to make a conjecture, predi...
 5.49: Verify that each equation is an identity. sin2 x  sin2 y = cos2 y ...
 5.50: Verify that each equation is an identity. 2 cos3 x  cos x = cos2 x...
 5.51: Verify that each equation is an identity. sin2 x 2  2 cos x = cos2...
 5.52: Verify that each equation is an identity. sin 2x sin x = 2 sec x
 5.53: Verify that each equation is an identity. 2 cos A  sec A = cos A ...
 5.54: Verify that each equation is an identity. 2 tan B sin 2B = sec2 B
 5.55: Verify that each equation is an identity. 1 + tan2 a = 2 tan a csc 2a
 5.56: Verify that each equation is an identity. 2 cot x tan 2x = csc2 x  2
 5.57: Verify that each equation is an identity. tan u sin 2u = 2  2 cos2 u
 5.58: Verify that each equation is an identity. csc A sin 2A  sec A = co...
 5.59: Verify that each equation is an identity. 2 tan x csc 2x  tan2 x = 1
 5.60: Verify that each equation is an identity. 2 cos2 u  1 = 1  tan2 u...
 5.61: Verify that each equation is an identity. tan u cos2 u = 2 tan u co...
 5.62: Verify that each equation is an identity. sec2 a  1 = sec 2a  1 s...
 5.63: Verify that each equation is an identity. sin2 x  cos2 x csc x = 2...
 5.64: Verify that each equation is an identity. sin3 u = sin u  cos2 u s...
 5.65: Verify that each equation is an identity. tan 4u = 2 tan 2u 2  sec...
 5.66: Verify that each equation is an identity. 2 cos2 x 2 tan x = tan x ...
 5.67: Verify that each equation is an identity. tan a x 2 + p 4 b = sec x...
 5.68: Verify that each equation is an identity. 1 2 cot x 2  1 2 tan x 2...
 5.69: Verify that each equation is an identity.  cot x 2 = sin 2x + sin ...
 5.70: Verify that each equation is an identity. sin 3t + sin 2t sin 3t  ...
 5.71: Distance Traveled by an Object The distance D of an object thrown (...
 5.72: Amperage, Wattage, and Voltage Suppose that for an electric heater,...
Solutions for Chapter 5: Trigonometric Identities
Full solutions for Trigonometry  11th Edition
ISBN: 9780134217437
Solutions for Chapter 5: Trigonometric Identities
Get Full SolutionsSince 72 problems in chapter 5: Trigonometric Identities have been answered, more than 9643 students have viewed full stepbystep solutions from this chapter. Chapter 5: Trigonometric Identities includes 72 full stepbystep solutions. Trigonometry was written by and is associated to the ISBN: 9780134217437. This textbook survival guide was created for the textbook: Trigonometry, edition: 11. This expansive textbook survival guide covers the following chapters and their solutions.

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

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.

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

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

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

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

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 .

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.

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.

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.

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

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

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.

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

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!

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

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

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

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

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.
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