 5.1: In Exercises 1 and 2, estimate the number of degrees in the angle. ...
 5.2: In Exercises 1 and 2, estimate the number of degrees in the angle. ...
 5.3: In Exercises 36, (a) sketch the angle in standard position, (b) det...
 5.4: In Exercises 36, (a) sketch the angle in standard position, (b) det...
 5.5: In Exercises 36, (a) sketch the angle in standard position, (b) det...
 5.6: In Exercises 36, (a) sketch the angle in standard position, (b) det...
 5.7: In Exercises 710, use the angleconversion capabilities of a graphi...
 5.8: In Exercises 710, use the angleconversion capabilities of a graphi...
 5.9: In Exercises 710, use the angleconversion capabilities of a graphi...
 5.10: In Exercises 710, use the angleconversion capabilities of a graphi...
 5.11: In Exercises 1114, use the angleconversion capabilities of a graph...
 5.12: In Exercises 1114, use the angleconversion capabilities of a graph...
 5.13: In Exercises 1114, use the angleconversion capabilities of a graph...
 5.14: In Exercises 1114, use the angleconversion capabilities of a graph...
 5.15: In Exercises 1518, find (if possible) the complement and supplement...
 5.16: In Exercises 1518, find (if possible) the complement and supplement...
 5.17: In Exercises 1518, find (if possible) the complement and supplement...
 5.18: In Exercises 1518, find (if possible) the complement and supplement...
 5.19: In Exercises 1922, (a) sketch the angle in standard position, (b) d...
 5.20: In Exercises 1922, (a) sketch the angle in standard position, (b) d...
 5.21: In Exercises 1922, (a) sketch the angle in standard position, (b) d...
 5.22: In Exercises 1922, (a) sketch the angle in standard position, (b) d...
 5.23: In Exercises 2326, convert the angle measure from degrees to radian...
 5.24: In Exercises 2326, convert the angle measure from degrees to radian...
 5.25: In Exercises 2326, convert the angle measure from degrees to radian...
 5.26: In Exercises 2326, convert the angle measure from degrees to radian...
 5.27: In Exercises 2730, convert the angle measure from radians to degree...
 5.28: In Exercises 2730, convert the angle measure from radians to degree...
 5.29: In Exercises 2730, convert the angle measure from radians to degree...
 5.30: In Exercises 2730, convert the angle measure from radians to degree...
 5.31: In Exercises 3134, find (if possible) the complement and supplement...
 5.32: In Exercises 3134, find (if possible) the complement and supplement...
 5.33: In Exercises 3134, find (if possible) the complement and supplement...
 5.34: In Exercises 3134, find (if possible) the complement and supplement...
 5.35: In Exercises 35 and 36, find the angle in radians. 35. 5 ft 6 ft
 5.36: In Exercises 35 and 36, find the angle in radians. 36. 12 in. 31 in.
 5.37: Find the length of the arc on a circle with a radius of 20 meters i...
 5.38: Find the length of the arc on a circle with a radius of 15 centimet...
 5.39: The radius of a compact disc is 6 centimeters. Find the linear spee...
 5.40: A car is moving at a rate of 28 miles per hour, and the diameter of...
 5.41: In Exercises 41 48, find the exact values of the six trigonometric ...
 5.42: In Exercises 41 48, find the exact values of the six trigonometric ...
 5.43: In Exercises 41 48, find the exact values of the six trigonometric ...
 5.44: In Exercises 41 48, find the exact values of the six trigonometric ...
 5.45: In Exercises 41 48, find the exact values of the six trigonometric ...
 5.46: In Exercises 41 48, find the exact values of the six trigonometric ...
 5.47: In Exercises 41 48, find the exact values of the six trigonometric ...
 5.48: In Exercises 41 48, find the exact values of the six trigonometric ...
 5.49: In Exercises 4952, use a calculator to evaluate each function. Roun...
 5.50: In Exercises 4952, use a calculator to evaluate each function. Roun...
 5.51: In Exercises 4952, use a calculator to evaluate each function. Roun...
 5.52: In Exercises 4952, use a calculator to evaluate each function. Roun...
 5.53: In Exercises 53 and 54, use trigonometric identities to transform o...
 5.54: In Exercises 53 and 54, use trigonometric identities to transform o...
 5.55: A surveyor is trying to determine the width of a river. From point ...
 5.56: An escalator 152 feet in length rises to a platform and makes a 30 ...
 5.57: In Exercises 5762, the point is on the terminal side of an angle in...
 5.58: In Exercises 5762, the point is on the terminal side of an angle in...
 5.59: In Exercises 5762, the point is on the terminal side of an angle in...
 5.60: In Exercises 5762, the point is on the terminal side of an angle in...
 5.61: In Exercises 5762, the point is on the terminal side of an angle in...
 5.62: In Exercises 5762, the point is on the terminal side of an angle in...
 5.63: In Exercises 6366, find the values of the other five trigonometric ...
 5.64: In Exercises 6366, find the values of the other five trigonometric ...
 5.65: In Exercises 6366, find the values of the other five trigonometric ...
 5.66: In Exercises 6366, find the values of the other five trigonometric ...
 5.67: In Exercises 6774, find the reference angle . Sketch in standard po...
 5.68: In Exercises 6774, find the reference angle . Sketch in standard po...
 5.69: In Exercises 6774, find the reference angle . Sketch in standard po...
 5.70: In Exercises 6774, find the reference angle . Sketch in standard po...
 5.71: In Exercises 6774, find the reference angle . Sketch in standard po...
 5.72: In Exercises 6774, find the reference angle . Sketch in standard po...
 5.73: In Exercises 6774, find the reference angle . Sketch in standard po...
 5.74: In Exercises 6774, find the reference angle . Sketch in standard po...
 5.75: In Exercises 7582, evaluate the sine, cosine, and tangent of the an...
 5.76: In Exercises 7582, evaluate the sine, cosine, and tangent of the an...
 5.77: In Exercises 7582, evaluate the sine, cosine, and tangent of the an...
 5.78: In Exercises 7582, evaluate the sine, cosine, and tangent of the an...
 5.79: In Exercises 7582, evaluate the sine, cosine, and tangent of the an...
 5.80: In Exercises 7582, evaluate the sine, cosine, and tangent of the an...
 5.81: In Exercises 7582, evaluate the sine, cosine, and tangent of the an...
 5.82: In Exercises 7582, evaluate the sine, cosine, and tangent of the an...
 5.83: In Exercises 83 86, use a calculator to evaluate the trigonometric ...
 5.84: In Exercises 83 86, use a calculator to evaluate the trigonometric ...
 5.85: In Exercises 83 86, use a calculator to evaluate the trigonometric ...
 5.86: In Exercises 83 86, use a calculator to evaluate the trigonometric ...
 5.87: In Exercises 87 90, sketch the graph of the function. 87. f(x) = 6 ...
 5.88: In Exercises 87 90, sketch the graph of the function. 88. f(x) = 7 ...
 5.89: In Exercises 87 90, sketch the graph of the function. 89. f(x) = 1 ...
 5.90: In Exercises 87 90, sketch the graph of the function. 90. f(x) = 3 ...
 5.91: In Exercises 91 94, find the period and amplitude. 91. 6 6 2
 5.92: In Exercises 91 94, find the period and amplitude. 92. y = 5 cos x ...
 5.93: In Exercises 91 94, find the period and amplitude. 93. 94. 2 2 4 4 93.
 5.94: In Exercises 91 94, find the period and amplitude. 94. y = 3.4 sin ...
 5.95: In Exercises 95 106, sketch the graph of the function. (Include two...
 5.96: In Exercises 95 106, sketch the graph of the function. (Include two...
 5.97: In Exercises 95 106, sketch the graph of the function. (Include two...
 5.98: In Exercises 95 106, sketch the graph of the function. (Include two...
 5.99: In Exercises 95 106, sketch the graph of the function. (Include two...
 5.100: In Exercises 95 106, sketch the graph of the function. (Include two...
 5.101: In Exercises 95 106, sketch the graph of the function. (Include two...
 5.102: In Exercises 95 106, sketch the graph of the function. (Include two...
 5.103: In Exercises 95 106, sketch the graph of the function. (Include two...
 5.104: In Exercises 95 106, sketch the graph of the function. (Include two...
 5.105: In Exercises 95 106, sketch the graph of the function. (Include two...
 5.106: In Exercises 95 106, sketch the graph of the function. (Include two...
 5.107: In Exercises 107110, find a, b, and c for the function f(x) = a cos...
 5.108: In Exercises 107110, find a, b, and c for the function f(x) = a cos...
 5.109: In Exercises 107110, find a, b, and c for the function f(x) = a cos...
 5.110: In Exercises 107110, find a, b, and c for the function f(x) = a cos...
 5.111: In Exercises 111 and 112, use a graphing utility to graph the sales...
 5.112: In Exercises 111 and 112, use a graphing utility to graph the sales...
 5.113: In Exercises 113126, sketch the graph of the function. (Include two...
 5.114: In Exercises 113126, sketch the graph of the function. (Include two...
 5.115: In Exercises 113126, sketch the graph of the function. (Include two...
 5.116: In Exercises 113126, sketch the graph of the function. (Include two...
 5.117: In Exercises 113126, sketch the graph of the function. (Include two...
 5.118: In Exercises 113126, sketch the graph of the function. (Include two...
 5.119: In Exercises 113126, sketch the graph of the function. (Include two...
 5.120: In Exercises 113126, sketch the graph of the function. (Include two...
 5.121: In Exercises 113126, sketch the graph of the function. (Include two...
 5.122: In Exercises 113126, sketch the graph of the function. (Include two...
 5.123: In Exercises 113126, sketch the graph of the function. (Include two...
 5.124: In Exercises 113126, sketch the graph of the function. (Include two...
 5.125: In Exercises 113126, sketch the graph of the function. (Include two...
 5.126: In Exercises 113126, sketch the graph of the function. (Include two...
 5.127: In Exercises 127 134, use a graphing utility to graph the function....
 5.128: In Exercises 127 134, use a graphing utility to graph the function....
 5.129: In Exercises 127 134, use a graphing utility to graph the function....
 5.130: In Exercises 127 134, use a graphing utility to graph the function....
 5.131: In Exercises 127 134, use a graphing utility to graph the function....
 5.132: In Exercises 127 134, use a graphing utility to graph the function....
 5.133: In Exercises 127 134, use a graphing utility to graph the function....
 5.134: In Exercises 127 134, use a graphing utility to graph the function....
 5.135: In Exercises 135 138, use a graphing utility to graph the function ...
 5.136: In Exercises 135 138, use a graphing utility to graph the function ...
 5.137: In Exercises 135 138, use a graphing utility to graph the function ...
 5.138: In Exercises 135 138, use a graphing utility to graph the function ...
 5.139: In Exercises 139 142, find the exact value of each expression witho...
 5.140: In Exercises 139 142, find the exact value of each expression witho...
 5.141: In Exercises 139 142, find the exact value of each expression witho...
 5.142: In Exercises 139 142, find the exact value of each expression witho...
 5.143: In Exercises 143 150, use a calculator to approximate the value of ...
 5.144: In Exercises 143 150, use a calculator to approximate the value of ...
 5.145: In Exercises 143 150, use a calculator to approximate the value of ...
 5.146: In Exercises 143 150, use a calculator to approximate the value of ...
 5.147: In Exercises 143 150, use a calculator to approximate the value of ...
 5.148: In Exercises 143 150, use a calculator to approximate the value of ...
 5.149: In Exercises 143 150, use a calculator to approximate the value of ...
 5.150: In Exercises 143 150, use a calculator to approximate the value of ...
 5.151: In Exercises 151 and 152, use an inverse trigonometric function to ...
 5.152: In Exercises 151 and 152, use an inverse trigonometric function to ...
 5.153: In Exercises 153 156, write an algebraic expression that is equival...
 5.154: In Exercises 153 156, write an algebraic expression that is equival...
 5.155: In Exercises 153 156, write an algebraic expression that is equival...
 5.156: In Exercises 153 156, write an algebraic expression that is equival...
 5.157: The height of a radio transmission tower is 70 meters, and it casts...
 5.158: An observer 2.5 miles from the launch pad of a space shuttle launch...
 5.159: A train travels 3.5 kilometers on a straight track with a grade of ...
 5.160: A road sign at the top of a mountain indicates that for the next 4 ...
 5.161: A passenger in an airplane flying at an altitude of 37,000 feet see...
 5.162: From city A to city B, a plane flies 650 miles at a bearing of 48. ...
 5.163: A buoy oscillates in simple harmonic motion as waves go past. The b...
 5.164: Your fishing bobber is oscillating in simple harmonic motion caused...
 5.165: In Exercises 165166, determine whether the statement is true or fal...
 5.166: In Exercises 165166, determine whether the statement is true or fal...
 5.167: You can use the cotangent function to model simple harmonic motion
 5.168: The sine of any nonacute angle is equal to the sine of the referenc...
 5.169: A 3000pound automobile is negotiating a circular interchange of ra...
 5.170: Using calculus, it can be shown that the secant function can be app...
 5.171: Using calculus, it can be shown that the arctangent function can be...
Solutions for Chapter 5: Trigonometric Functions
Full solutions for Algebra and Trigonometry: Real Mathematics, Real People  7th Edition
ISBN: 9781305071735
Solutions for Chapter 5: Trigonometric Functions
Get Full SolutionsChapter 5: Trigonometric Functions includes 171 full stepbystep solutions. Algebra and Trigonometry: Real Mathematics, Real People was written by and is associated to the ISBN: 9781305071735. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Algebra and Trigonometry: Real Mathematics, Real People, edition: 7. Since 171 problems in chapter 5: Trigonometric Functions have been answered, more than 60870 students have viewed full stepbystep solutions from this chapter.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

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.

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.)

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).

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.

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

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].

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

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.

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

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

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

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.

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

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

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.