 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 20301 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.

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.

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.

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.

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.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

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 combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

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).

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.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

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.

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

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

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.

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.

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.