 4.4.4.4.1: In Exercises 16, let be an angle in standard position with a point ...
 4.4.4.4.2: In Exercises 16, let be an angle in standard position with a point ...
 4.4.4.4.3: In Exercises 16, let be an angle in standard position with a point ...
 4.4.4.4.4: In Exercises 16, let be an angle in standard position with a point ...
 4.4.4.4.5: In Exercises 16, let be an angle in standard position with a point ...
 4.4.4.4.6: In Exercises 16, let be an angle in standard position with a point ...
 4.4.4.4.7: The acute positive angle that is formed by the terminal side of the...
 4.4.4.4.8: In Exercises 512, the point is on the terminal side of an angle in ...
 4.4.4.4.9: In Exercises 512, the point is on the terminal side of an angle in ...
 4.4.4.4.10: In Exercises 512, the point is on the terminal side of an angle in ...
 4.4.4.4.11: In Exercises 512, the point is on the terminal side of an angle in ...
 4.4.4.4.12: In Exercises 512, the point is on the terminal side of an angle in ...
 4.4.4.4.13: In Exercises 1316, state the quadrant in which lies.
 4.4.4.4.14: In Exercises 1316, state the quadrant in which lies.
 4.4.4.4.15: In Exercises 1316, state the quadrant in which lies.
 4.4.4.4.16: In Exercises 1316, state the quadrant in which lies.
 4.4.4.4.17: In Exercises 1724, find the values of the six trigonometric functio...
 4.4.4.4.18: In Exercises 1724, find the values of the six trigonometric functio...
 4.4.4.4.19: In Exercises 1724, find the values of the six trigonometric functio...
 4.4.4.4.20: In Exercises 1724, find the values of the six trigonometric functio...
 4.4.4.4.21: In Exercises 1724, find the values of the six trigonometric functio...
 4.4.4.4.22: In Exercises 1724, find the values of the six trigonometric functio...
 4.4.4.4.23: In Exercises 1724, find the values of the six trigonometric functio...
 4.4.4.4.24: In Exercises 1724, find the values of the six trigonometric functio...
 4.4.4.4.25: In Exercises 2528, the terminal side of lies on the given line in t...
 4.4.4.4.26: In Exercises 2528, the terminal side of lies on the given line in t...
 4.4.4.4.27: In Exercises 2528, the terminal side of lies on the given line in t...
 4.4.4.4.28: In Exercises 2528, the terminal side of lies on the given line in t...
 4.4.4.4.29: In Exercises 2936, evaluate the trigonometric function of the quadr...
 4.4.4.4.30: In Exercises 2936, evaluate the trigonometric function of the quadr...
 4.4.4.4.31: In Exercises 2936, evaluate the trigonometric function of the quadr...
 4.4.4.4.32: In Exercises 2936, evaluate the trigonometric function of the quadr...
 4.4.4.4.33: In Exercises 2936, evaluate the trigonometric function of the quadr...
 4.4.4.4.34: In Exercises 2936, evaluate the trigonometric function of the quadr...
 4.4.4.4.35: In Exercises 2936, evaluate the trigonometric function of the quadr...
 4.4.4.4.36: In Exercises 2936, evaluate the trigonometric function of the quadr...
 4.4.4.4.37: In Exercises 37 44, find the reference angle for the special angle ...
 4.4.4.4.38: In Exercises 37 44, find the reference angle for the special angle ...
 4.4.4.4.39: In Exercises 37 44, find the reference angle for the special angle ...
 4.4.4.4.40: In Exercises 37 44, find the reference angle for the special angle ...
 4.4.4.4.41: In Exercises 37 44, find the reference angle for the special angle ...
 4.4.4.4.42: In Exercises 37 44, find the reference angle for the special angle ...
 4.4.4.4.43: In Exercises 37 44, find the reference angle for the special angle ...
 4.4.4.4.44: In Exercises 37 44, find the reference angle for the special angle ...
 4.4.4.4.45: In Exercises 4552, find the reference angle and sketch and in stand...
 4.4.4.4.46: In Exercises 4552, find the reference angle and sketch and in stand...
 4.4.4.4.47: In Exercises 4552, find the reference angle and sketch and in stand...
 4.4.4.4.48: In Exercises 4552, find the reference angle and sketch and in stand...
 4.4.4.4.49: In Exercises 4552, find the reference angle and sketch and in stand...
 4.4.4.4.50: In Exercises 4552, find the reference angle and sketch and in stand...
 4.4.4.4.51: In Exercises 4552, find the reference angle and sketch and in stand...
 4.4.4.4.52: In Exercises 4552, find the reference angle and sketch and in stand...
 4.4.4.4.53: In Exercises 5364, evaluate the sine, cosine, and tangent of the an...
 4.4.4.4.54: In Exercises 5364, evaluate the sine, cosine, and tangent of the an...
 4.4.4.4.55: In Exercises 5364, evaluate the sine, cosine, and tangent of the an...
 4.4.4.4.56: In Exercises 5364, evaluate the sine, cosine, and tangent of the an...
 4.4.4.4.57: In Exercises 5364, evaluate the sine, cosine, and tangent of the an...
 4.4.4.4.58: In Exercises 5364, evaluate the sine, cosine, and tangent of the an...
 4.4.4.4.59: In Exercises 5364, evaluate the sine, cosine, and tangent of the an...
 4.4.4.4.60: In Exercises 5364, evaluate the sine, cosine, and tangent of the an...
 4.4.4.4.61: In Exercises 5364, evaluate the sine, cosine, and tangent of the an...
 4.4.4.4.62: In Exercises 5364, evaluate the sine, cosine, and tangent of the an...
 4.4.4.4.63: In Exercises 5364, evaluate the sine, cosine, and tangent of the an...
 4.4.4.4.64: In Exercises 5364, evaluate the sine, cosine, and tangent of the an...
 4.4.4.4.65: In Exercises 6570, find the indicated trigonometric value in the sp...
 4.4.4.4.66: In Exercises 6570, find the indicated trigonometric value in the sp...
 4.4.4.4.67: In Exercises 6570, find the indicated trigonometric value in the sp...
 4.4.4.4.68: In Exercises 6570, find the indicated trigonometric value in the sp...
 4.4.4.4.69: In Exercises 6570, find the indicated trigonometric value in the sp...
 4.4.4.4.70: In Exercises 6570, find the indicated trigonometric value in the sp...
 4.4.4.4.71: In Exercises 7176, use the given value and the trigonometric identi...
 4.4.4.4.72: In Exercises 7176, use the given value and the trigonometric identi...
 4.4.4.4.73: In Exercises 7176, use the given value and the trigonometric identi...
 4.4.4.4.74: In Exercises 7176, use the given value and the trigonometric identi...
 4.4.4.4.75: In Exercises 7176, use the given value and the trigonometric identi...
 4.4.4.4.76: In Exercises 7176, use the given value and the trigonometric identi...
 4.4.4.4.77: In Exercises 7788, use a calculator to evaluate the trigonometric f...
 4.4.4.4.78: In Exercises 7788, use a calculator to evaluate the trigonometric f...
 4.4.4.4.79: In Exercises 7788, use a calculator to evaluate the trigonometric f...
 4.4.4.4.80: In Exercises 7788, use a calculator to evaluate the trigonometric f...
 4.4.4.4.81: In Exercises 7788, use a calculator to evaluate the trigonometric f...
 4.4.4.4.82: In Exercises 7788, use a calculator to evaluate the trigonometric f...
 4.4.4.4.83: In Exercises 7788, use a calculator to evaluate the trigonometric f...
 4.4.4.4.84: In Exercises 7788, use a calculator to evaluate the trigonometric f...
 4.4.4.4.85: In Exercises 7788, use a calculator to evaluate the trigonometric f...
 4.4.4.4.86: In Exercises 7788, use a calculator to evaluate the trigonometric f...
 4.4.4.4.87: In Exercises 7788, use a calculator to evaluate the trigonometric f...
 4.4.4.4.88: In Exercises 7788, use a calculator to evaluate the trigonometric f...
 4.4.4.4.89: In Exercises 8994, find two solutions of the equation. Give your an...
 4.4.4.4.90: In Exercises 8994, find two solutions of the equation. Give your an...
 4.4.4.4.91: In Exercises 8994, find two solutions of the equation. Give your an...
 4.4.4.4.92: In Exercises 8994, find two solutions of the equation. Give your an...
 4.4.4.4.93: In Exercises 8994, find two solutions of the equation. Give your an...
 4.4.4.4.94: In Exercises 8994, find two solutions of the equation. Give your an...
 4.4.4.4.95: In Exercises 95108, find the exact value of each function for the g...
 4.4.4.4.96: In Exercises 95108, find the exact value of each function for the g...
 4.4.4.4.97: In Exercises 95108, find the exact value of each function for the g...
 4.4.4.4.98: In Exercises 95108, find the exact value of each function for the g...
 4.4.4.4.99: In Exercises 95108, find the exact value of each function for the g...
 4.4.4.4.100: In Exercises 95108, find the exact value of each function for the g...
 4.4.4.4.101: In Exercises 95108, find the exact value of each function for the g...
 4.4.4.4.102: In Exercises 95108, find the exact value of each function for the g...
 4.4.4.4.103: In Exercises 95108, find the exact value of each function for the g...
 4.4.4.4.104: In Exercises 95108, find the exact value of each function for the g...
 4.4.4.4.105: In Exercises 95108, find the exact value of each function for the g...
 4.4.4.4.106: In Exercises 95108, find the exact value of each function for the g...
 4.4.4.4.107: In Exercises 95108, find the exact value of each function for the g...
 4.4.4.4.108: In Exercises 95108, find the exact value of each function for the g...
 4.4.4.4.109: Meteorology The monthly normal temperatures (in degrees Fahrenheit)...
 4.4.4.4.110: Sales A company that produces water skis, which are seasonal produc...
 4.4.4.4.111: Distance An airplane flying at an altitude of 6 miles is on a fligh...
 4.4.4.4.112: Writing Consider an angle in standard position with centimeters, as...
 4.4.4.4.113: True or False? In Exercises 113 and 114, determine whether the stat...
 4.4.4.4.114: True or False? In Exercises 113 and 114, determine whether the stat...
 4.4.4.4.115: Conjecture (a) Use a graphing utility to complete the table. (b) Ma...
 4.4.4.4.116: Writing Create a table of the six trigonometric functions comparing...
 4.4.4.4.117: In Exercises 117126, solve the equation. Round your answer to three...
 4.4.4.4.118: In Exercises 117126, solve the equation. Round your answer to three...
 4.4.4.4.119: In Exercises 117126, solve the equation. Round your answer to three...
 4.4.4.4.120: In Exercises 117126, solve the equation. Round your answer to three...
 4.4.4.4.121: In Exercises 117126, solve the equation. Round your answer to three...
 4.4.4.4.122: In Exercises 117126, solve the equation. Round your answer to three...
 4.4.4.4.123: In Exercises 117126, solve the equation. Round your answer to three...
 4.4.4.4.124: In Exercises 117126, solve the equation. Round your answer to three...
 4.4.4.4.125: In Exercises 117126, solve the equation. Round your answer to three...
 4.4.4.4.126: In Exercises 117126, solve the equation. Round your answer to three...
Solutions for Chapter 4.4: Trigonometric Functions of Any Angle
Full solutions for Precalculus With Limits A Graphing Approach  5th Edition
ISBN: 9780618851522
Solutions for Chapter 4.4: Trigonometric Functions of Any Angle
Get Full SolutionsSince 126 problems in chapter 4.4: Trigonometric Functions of Any Angle have been answered, more than 36196 students have viewed full stepbystep solutions from this chapter. Chapter 4.4: Trigonometric Functions of Any Angle includes 126 full stepbystep solutions. Precalculus With Limits A Graphing Approach was written by and is associated to the ISBN: 9780618851522. This textbook survival guide was created for the textbook: Precalculus With Limits A Graphing Approach, edition: 5. This expansive textbook survival guide covers the following chapters and their solutions.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

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!

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.)

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

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

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.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

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.

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.

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.

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.

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.

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