 7.3.1: Exer. 14: Express as a cofunction of a complementary angle.
 7.3.2: Exer. 14: Express as a cofunction of a complementary angle.
 7.3.3: Exer. 14: Express as a cofunction of a complementary angle.
 7.3.4: Exer. 14: Express as a cofunction of a complementary angle.
 7.3.5: Exer. 510: Find the exact values.
 7.3.6: Exer. 510: Find the exact values.
 7.3.7: Exer. 510: Find the exact values.
 7.3.8: Exer. 510: Find the exact values.
 7.3.9: Exer. 510: Find the exact values.
 7.3.10: Exer. 510: Find the exact values.
 7.3.11: Exer. 1116: Express as a trigonometric function of one angle
 7.3.12: Exer. 1116: Express as a trigonometric function of one angle
 7.3.13: Exer. 1116: Express as a trigonometric function of one angle
 7.3.14: Exer. 1116: Express as a trigonometric function of one angle
 7.3.15: Exer. 1116: Express as a trigonometric function of one angle
 7.3.16: Exer. 1116: Express as a trigonometric function of one angle
 7.3.17: If and , find the exact value of
 7.3.18: If and , find the exact value of
 7.3.19: If and , find the exact value of
 7.3.20: If and , find the exact value of
 7.3.21: If and , find the exact value of
 7.3.22: If and , find the exact value of
 7.3.23: If and , find the exact value of
 7.3.24: If and , find the exact value of
 7.3.25: Exer. 2536: Verify the reduction formula.
 7.3.26: Exer. 2536: Verify the reduction formula.
 7.3.27: Exer. 2536: Verify the reduction formula.
 7.3.28: Exer. 2536: Verify the reduction formula.
 7.3.29: Exer. 2536: Verify the reduction formula.
 7.3.30: Exer. 2536: Verify the reduction formula.
 7.3.31: Exer. 2536: Verify the reduction formula.
 7.3.32: Exer. 2536: Verify the reduction formula.
 7.3.33: Exer. 2536: Verify the reduction formula.
 7.3.34: Exer. 2536: Verify the reduction formula.
 7.3.35: Exer. 2536: Verify the reduction formula.
 7.3.36: Exer. 2536: Verify the reduction formula.
 7.3.37: Exer. 3746: Verify the identity.
 7.3.38: Exer. 3746: Verify the identity.
 7.3.39: Exer. 3746: Verify the identity.
 7.3.40: Exer. 3746: Verify the identity.
 7.3.41: Exer. 3746: Verify the identity.
 7.3.42: Exer. 3746: Verify the identity.
 7.3.43: Exer. 3746: Verify the identity.
 7.3.44: Exer. 3746: Verify the identity.
 7.3.45: Exer. 3746: Verify the identity.
 7.3.46: Exer. 3746: Verify the identity.
 7.3.47: Express in terms of trigonometric functions of u, v, and w. (Hint: ...
 7.3.48: Express in terms of trigonometric functions of u, v, and w.
 7.3.49: Derive the formula .
 7.3.50: If and are complementary angles, show that
 7.3.51: Derive the subtraction formula for the sine function.
 7.3.52: Derive the subtraction formula for the tangent function.
 7.3.53: If , show that
 7.3.54: If , show that
 7.3.55: Exer. 5556: (a) Compare the decimal approximations of both sides of...
 7.3.56: Exer. 5556: (a) Compare the decimal approximations of both sides of...
 7.3.57: Exer. 5762: Use an addition or subtraction formula to find the solu...
 7.3.58: Exer. 5762: Use an addition or subtraction formula to find the solu...
 7.3.59: Exer. 5762: Use an addition or subtraction formula to find the solu...
 7.3.60: Exer. 5762: Use an addition or subtraction formula to find the solu...
 7.3.61: Exer. 5762: Use an addition or subtraction formula to find the solu...
 7.3.62: Exer. 5762: Use an addition or subtraction formula to find the solu...
 7.3.63: Exer. 6366: (a) Use the formula from Example 6 to express f in term...
 7.3.64: Exer. 6366: (a) Use the formula from Example 6 to express f in term...
 7.3.65: Exer. 6366: (a) Use the formula from Example 6 to express f in term...
 7.3.66: Exer. 6366: (a) Use the formula from Example 6 to express f in term...
 7.3.67: Exer. 6768: For certain applications in electrical engineering, the...
 7.3.68: Exer. 6768: For certain applications in electrical engineering, the...
 7.3.69: If a mass that is attached to a spring is raised feet and released ...
 7.3.70: Refer to Exercise 69. If and , find the initial velocities that res...
 7.3.71: If a tuning fork is struck and then held a certain distance from th...
 7.3.72: Refer to Exercise 71. Destructive interference occurs if the amplit...
 7.3.73: Refer to Exercise 71. When two tuning forks are struck, constructiv...
Solutions for Chapter 7.3: The Addition and Subtr action For mulas
Full solutions for Algebra and Trigonometry with Analytic Geometry  12th Edition
ISBN: 9780495559719
Solutions for Chapter 7.3: The Addition and Subtr action For mulas
Get Full SolutionsAlgebra and Trigonometry with Analytic Geometry was written by and is associated to the ISBN: 9780495559719. Chapter 7.3: The Addition and Subtr action For mulas includes 73 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 73 problems in chapter 7.3: The Addition and Subtr action For mulas have been answered, more than 33456 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Algebra and Trigonometry with Analytic Geometry, edition: 12.

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).

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

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

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

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

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.

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.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

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

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

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

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).

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

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 addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).