 4.4.1: In Exercises 1 and 2, estimate the angle to the nearest onehalf ra...
 4.4.2: In Exercises 1 and 2, estimate the angle to the nearest onehalf ra...
 4.4.3: In Exercises 36, (a) sketch the angle in standard position, (b) det...
 4.4.4: In Exercises 36, (a) sketch the angle in standard position, (b) det...
 4.4.5: In Exercises 36, (a) sketch the angle in standard position, (b) det...
 4.4.6: In Exercises 36, (a) sketch the angle in standard position, (b) det...
 4.4.7: In Exercises 710, find (if possible) the complement and supplement ...
 4.4.8: In Exercises 710, find (if possible) the complement and supplement ...
 4.4.9: In Exercises 710, find (if possible) the complement and supplement ...
 4.4.10: In Exercises 710, find (if possible) the complement and supplement ...
 4.4.11: In Exercises 1114, (a) sketch the angle in standard position, (b) d...
 4.4.12: In Exercises 1114, (a) sketch the angle in standard position, (b) d...
 4.4.13: In Exercises 1114, (a) sketch the angle in standard position, (b) d...
 4.4.14: In Exercises 1114, (a) sketch the angle in standard position, (b) d...
 4.4.15: In Exercises 1518, find (if possible) the complement and supplement...
 4.4.16: In Exercises 1518, find (if possible) the complement and supplement...
 4.4.17: In Exercises 1518, find (if possible) the complement and supplement...
 4.4.18: In Exercises 1518, find (if possible) the complement and supplement...
 4.4.19: In Exercises 1922, use the angleconversion capabilities of a graph...
 4.4.20: In Exercises 1922, use the angleconversion capabilities of a graph...
 4.4.21: In Exercises 1922, use the angleconversion capabilities of a graph...
 4.4.22: In Exercises 1922, use the angleconversion capabilities of a graph...
 4.4.23: In Exercises 2326, use the angleconversion capabilities of a graph...
 4.4.24: In Exercises 2326, use the angleconversion capabilities of a graph...
 4.4.25: In Exercises 2326, use the angleconversion capabilities of a graph...
 4.4.26: In Exercises 2326, use the angleconversion capabilities of a graph...
 4.4.27: In Exercises 2730, convert the angle measure from degrees to radian...
 4.4.28: In Exercises 2730, convert the angle measure from degrees to radian...
 4.4.29: In Exercises 2730, convert the angle measure from degrees to radian...
 4.4.30: In Exercises 2730, convert the angle measure from degrees to radian...
 4.4.31: In Exercises 3134, convert the angle measure from radians to degree...
 4.4.32: In Exercises 3134, convert the angle measure from radians to degree...
 4.4.33: In Exercises 3134, convert the angle measure from radians to degree...
 4.4.34: In Exercises 3134, convert the angle measure from radians to degree...
 4.4.35: Find the radian measure of the central angle of a circle with a rad...
 4.4.36: Find the radian measure of the central angle of a circle with a rad...
 4.4.37: Find the length of the arc on a circle with a radius of 20 meters i...
 4.4.38: Find the length of the arc on a circle with a radius of 15 centimet...
 4.4.39: Music The radius of a compact disc is 6 centimeters. Find the linea...
 4.4.40: Angular Speed A car is moving at a rate of 28 miles per hour, and t...
 4.4.41: In Exercises 4148, find the point on the unit circle that correspon...
 4.4.42: In Exercises 4148, find the point on the unit circle that correspon...
 4.4.43: In Exercises 4148, find the point on the unit circle that correspon...
 4.4.44: In Exercises 4148, find the point on the unit circle that correspon...
 4.4.45: In Exercises 4148, find the point on the unit circle that correspon...
 4.4.46: In Exercises 4148, find the point on the unit circle that correspon...
 4.4.47: In Exercises 4148, find the point on the unit circle that correspon...
 4.4.48: In Exercises 4148, find the point on the unit circle that correspon...
 4.4.49: In Exercises 4956, evaluate (if possible) the six trigonometric fun...
 4.4.50: In Exercises 4956, evaluate (if possible) the six trigonometric fun...
 4.4.51: In Exercises 4956, evaluate (if possible) the six trigonometric fun...
 4.4.52: In Exercises 4956, evaluate (if possible) the six trigonometric fun...
 4.4.53: In Exercises 4956, evaluate (if possible) the six trigonometric fun...
 4.4.54: In Exercises 4956, evaluate (if possible) the six trigonometric fun...
 4.4.55: In Exercises 4956, evaluate (if possible) the six trigonometric fun...
 4.4.56: In Exercises 4956, evaluate (if possible) the six trigonometric fun...
 4.4.57: In Exercises 5760, evaluate the trigonometric function using its pe...
 4.4.58: In Exercises 5760, evaluate the trigonometric function using its pe...
 4.4.59: In Exercises 5760, evaluate the trigonometric function using its pe...
 4.4.60: In Exercises 5760, evaluate the trigonometric function using its pe...
 4.4.61: In Exercises 6164, use the value of the trigonometric function to e...
 4.4.62: In Exercises 6164, use the value of the trigonometric function to e...
 4.4.63: In Exercises 6164, use the value of the trigonometric function to e...
 4.4.64: In Exercises 6164, use the value of the trigonometric function to e...
 4.4.65: In Exercises 6568, use a calculator to evaluate the expression. Rou...
 4.4.66: In Exercises 6568, use a calculator to evaluate the expression. Rou...
 4.4.67: In Exercises 6568, use a calculator to evaluate the expression. Rou...
 4.4.68: In Exercises 6568, use a calculator to evaluate the expression. Rou...
 4.4.69: In Exercises 6972, find the exact values of the six trigonometric f...
 4.4.70: In Exercises 6972, find the exact values of the six trigonometric f...
 4.4.71: In Exercises 6972, find the exact values of the six trigonometric f...
 4.4.72: In Exercises 6972, find the exact values of the six trigonometric f...
 4.4.73: In Exercises 73 and 74, use trigonometric identities to transform o...
 4.4.74: In Exercises 73 and 74, use trigonometric identities to transform o...
 4.4.75: In Exercises 7578, use a calculator to evaluate each function. Roun...
 4.4.76: In Exercises 7578, use a calculator to evaluate each function. Roun...
 4.4.77: In Exercises 7578, use a calculator to evaluate each function. Roun...
 4.4.78: In Exercises 7578, use a calculator to evaluate each function. Roun...
 4.4.79: Width An engineer is trying to determine the width of a river (see ...
 4.4.80: Height An escalator 152 feet in length rises to a platform and make...
 4.4.81: In Exercises 8186, the point is on the terminal side of an angle in...
 4.4.82: In Exercises 8186, the point is on the terminal side of an angle in...
 4.4.83: In Exercises 8186, the point is on the terminal side of an angle in...
 4.4.84: In Exercises 8186, the point is on the terminal side of an angle in...
 4.4.85: In Exercises 8186, the point is on the terminal side of an angle in...
 4.4.86: In Exercises 8186, the point is on the terminal side of an angle in...
 4.4.87: In Exercises 8790, find the values of the other five trigonometric ...
 4.4.88: In Exercises 8790, find the values of the other five trigonometric ...
 4.4.89: In Exercises 8790, find the values of the other five trigonometric ...
 4.4.90: In Exercises 8790, find the values of the other five trigonometric ...
 4.4.91: In Exercises 9194, find the reference angle and sketch and in stand...
 4.4.92: In Exercises 9194, find the reference angle and sketch and in stand...
 4.4.93: In Exercises 9194, find the reference angle and sketch and in stand...
 4.4.94: In Exercises 9194, find the reference angle and sketch and in stand...
 4.4.95: In Exercises 95102, evaluate the sine, cosine, and tangent of the a...
 4.4.96: In Exercises 95102, evaluate the sine, cosine, and tangent of the a...
 4.4.97: In Exercises 95102, evaluate the sine, cosine, and tangent of the a...
 4.4.98: In Exercises 95102, evaluate the sine, cosine, and tangent of the a...
 4.4.99: In Exercises 95102, evaluate the sine, cosine, and tangent of the a...
 4.4.100: In Exercises 95102, evaluate the sine, cosine, and tangent of the a...
 4.4.101: In Exercises 95102, evaluate the sine, cosine, and tangent of the a...
 4.4.102: In Exercises 95102, evaluate the sine, cosine, and tangent of the a...
 4.4.103: In Exercises 103106, use a calculator to evaluate the trigonometric...
 4.4.104: In Exercises 103106, use a calculator to evaluate the trigonometric...
 4.4.105: In Exercises 103106, use a calculator to evaluate the trigonometric...
 4.4.106: In Exercises 103106, use a calculator to evaluate the trigonometric...
 4.4.107: In Exercises 107110, sketch the graph of the function.
 4.4.108: In Exercises 107110, sketch the graph of the function.
 4.4.109: In Exercises 107110, sketch the graph of the function.
 4.4.110: In Exercises 107110, sketch the graph of the function.
 4.4.111: In Exercises 111114, find the period and amplitude
 4.4.112: In Exercises 111114, find the period and amplitude
 4.4.113: In Exercises 111114, find the period and amplitude
 4.4.114: In Exercises 111114, find the period and amplitude
 4.4.115: In Exercises 115126, sketch the graph of the function. (Include two...
 4.4.116: In Exercises 115126, sketch the graph of the function. (Include two...
 4.4.117: In Exercises 115126, sketch the graph of the function. (Include two...
 4.4.118: In Exercises 115126, sketch the graph of the function. (Include two...
 4.4.119: In Exercises 115126, sketch the graph of the function. (Include two...
 4.4.120: In Exercises 115126, sketch the graph of the function. (Include two...
 4.4.121: In Exercises 115126, sketch the graph of the function. (Include two...
 4.4.122: In Exercises 115126, sketch the graph of the function. (Include two...
 4.4.123: In Exercises 115126, sketch the graph of the function. (Include two...
 4.4.124: In Exercises 115126, sketch the graph of the function. (Include two...
 4.4.125: In Exercises 115126, sketch the graph of the function. (Include two...
 4.4.126: In Exercises 115126, sketch the graph of the function. (Include two...
 4.4.127: Graphical Reasoning In Exercises 127130, find and for the function ...
 4.4.128: Graphical Reasoning In Exercises 127130, find and for the function ...
 4.4.129: Graphical Reasoning In Exercises 127130, find and for the function ...
 4.4.130: Graphical Reasoning In Exercises 127130, find and for the function ...
 4.4.131: Sales In Exercises 131 and 132, use a graphing utility to graph the...
 4.4.132: Sales In Exercises 131 and 132, use a graphing utility to graph the...
 4.4.133: In Exercises 133146, sketch the graph of the function. (Include two...
 4.4.134: In Exercises 133146, sketch the graph of the function. (Include two...
 4.4.135: In Exercises 133146, sketch the graph of the function. (Include two...
 4.4.136: In Exercises 133146, sketch the graph of the function. (Include two...
 4.4.137: In Exercises 133146, sketch the graph of the function. (Include two...
 4.4.138: In Exercises 133146, sketch the graph of the function. (Include two...
 4.4.139: In Exercises 133146, sketch the graph of the function. (Include two...
 4.4.140: In Exercises 133146, sketch the graph of the function. (Include two...
 4.4.141: In Exercises 133146, sketch the graph of the function. (Include two...
 4.4.142: In Exercises 133146, sketch the graph of the function. (Include two...
 4.4.143: In Exercises 133146, sketch the graph of the function. (Include two...
 4.4.144: In Exercises 133146, sketch the graph of the function. (Include two...
 4.4.145: In Exercises 133146, sketch the graph of the function. (Include two...
 4.4.146: In Exercises 133146, sketch the graph of the function. (Include two...
 4.4.147: In Exercises 147154, use a graphing utility to graph the function. ...
 4.4.148: In Exercises 147154, use a graphing utility to graph the function. ...
 4.4.149: In Exercises 147154, use a graphing utility to graph the function. ...
 4.4.150: In Exercises 147154, use a graphing utility to graph the function. ...
 4.4.151: In Exercises 147154, use a graphing utility to graph the function. ...
 4.4.152: In Exercises 147154, use a graphing utility to graph the function. ...
 4.4.153: In Exercises 147154, use a graphing utility to graph the function. ...
 4.4.154: In Exercises 147154, use a graphing utility to graph the function. ...
 4.4.155: In Exercises 155158, use a graphing utility to graph the function a...
 4.4.156: In Exercises 155158, use a graphing utility to graph the function a...
 4.4.157: In Exercises 155158, use a graphing utility to graph the function a...
 4.4.158: In Exercises 155158, use a graphing utility to graph the function a...
 4.4.159: In Exercises 159162, find the exact value of each expression withou...
 4.4.160: In Exercises 159162, find the exact value of each expression withou...
 4.4.161: In Exercises 159162, find the exact value of each expression withou...
 4.4.162: In Exercises 159162, find the exact value of each expression withou...
 4.4.163: In Exercises 163170, use a calculator to approximate the value of t...
 4.4.164: In Exercises 163170, use a calculator to approximate the value of t...
 4.4.165: In Exercises 163170, use a calculator to approximate the value of t...
 4.4.166: In Exercises 163170, use a calculator to approximate the value of t...
 4.4.167: In Exercises 163170, use a calculator to approximate the value of t...
 4.4.168: In Exercises 163170, use a calculator to approximate the value of t...
 4.4.169: In Exercises 163170, use a calculator to approximate the value of t...
 4.4.170: In Exercises 163170, use a calculator to approximate the value of t...
 4.4.171: In Exercises 171 and 172, use an inverse trigonometric function to ...
 4.4.172: In Exercises 171 and 172, use an inverse trigonometric function to ...
 4.4.173: In Exercises 173176, write an algebraic expression that is equivale...
 4.4.174: In Exercises 173176, write an algebraic expression that is equivale...
 4.4.175: In Exercises 173176, write an algebraic expression that is equivale...
 4.4.176: In Exercises 173176, write an algebraic expression that is equivale...
 4.4.177: Railroad Grade A train travels 3.5 kilometers on a straight track w...
 4.4.178: Mountain Descent A road sign at the top of a mountain indicates tha...
 4.4.179: Distance A passenger in an airplane flying at an altitude of 37,000...
 4.4.180: Distance From city A to city B, a plane flies 650 miles at a bearin...
 4.4.181: Wave Motion A buoy oscillates in simple harmonic motion as waves go...
 4.4.182: Wave Motion Your fishing bobber oscillates in simple harmonic motio...
 4.4.183: True or False? In Exercises 183 and 184, determine whether the stat...
 4.4.184: True or False? In Exercises 183 and 184, determine whether the stat...
 4.4.185: Numerical Analysis A 3000pound automobile is negotiating a circula...
 4.4.186: Approximation In calculus it can be shown that the arctangent funct...
 4.4.1: Consider an angle that measures radians. (a) Sketch the angle in st...
 4.4.2: A truck is moving at a rate of 90 kilometers per hour, and the diam...
 4.4.3: Find the exact values of the six trigonometric functions of the ang...
 4.4.4: Given that and is an acute angle, find the other five trigonometric...
 4.4.5: Determine the reference angle of the angle and sketch and in standa...
 4.4.6: Determine the quadrant in which lies if and
 4.4.7: Find two exact values of in degrees
 4.4.8: Use a calculator to approximate two values of in radians if Round y...
 4.4.9: Find the five remaining trigonometric functions of given that cos 3...
 4.4.10: In Exercises 1015, sketch the graph of the function. (Include two f...
 4.4.11: In Exercises 1015, sketch the graph of the function. (Include two f...
 4.4.12: In Exercises 1015, sketch the graph of the function. (Include two f...
 4.4.13: In Exercises 1015, sketch the graph of the function. (Include two f...
 4.4.14: In Exercises 1015, sketch the graph of the function. (Include two f...
 4.4.15: In Exercises 1015, sketch the graph of the function. (Include two f...
 4.4.16: In Exercises 16 and 17, use a graphing utility to graph the functio...
 4.4.17: In Exercises 16 and 17, use a graphing utility to graph the functio...
 4.4.18: Find and for the function such that the graph of matches the graph ...
 4.4.19: Find the exact value of without using a calculator.
 4.4.20: In Exercises 2022, use a graphing utility to graph the function.
 4.4.21: In Exercises 2022, use a graphing utility to graph the function.
 4.4.22: In Exercises 2022, use a graphing utility to graph the function.
 4.4.23: A plane is 160 miles north and 110 miles east of an airport. What b...
Solutions for Chapter 4: Trigonometric Functions
Full solutions for Precalculus With Limits A Graphing Approach  5th Edition
ISBN: 9780618851522
Solutions for Chapter 4: Trigonometric Functions
Get Full SolutionsThis textbook survival guide was created for the textbook: Precalculus With Limits A Graphing Approach, edition: 5. Chapter 4: Trigonometric Functions includes 209 full stepbystep solutions. Precalculus With Limits A Graphing Approach was written by and is associated to the ISBN: 9780618851522. Since 209 problems in chapter 4: Trigonometric Functions have been answered, more than 33077 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

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.

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

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

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.

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.

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

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.

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

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.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

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.

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

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).