 6.2.1: In Exercises 18, fill in the blank to complete the trigonometric id...
 6.2.2: In Exercises 18, fill in the blank to complete the trigonometric id...
 6.2.3: In Exercises 18, fill in the blank to complete the trigonometric id...
 6.2.4: In Exercises 18, fill in the blank to complete the trigonometric id...
 6.2.5: In Exercises 18, fill in the blank to complete the trigonometric id...
 6.2.6: In Exercises 18, fill in the blank to complete the trigonometric id...
 6.2.7: In Exercises 18, fill in the blank to complete the trigonometric id...
 6.2.8: In Exercises 18, fill in the blank to complete the trigonometric id...
 6.2.9: Is a graphical solution sufficient to verify a trigonometric identity?
 6.2.10: Is a conditional equation true for all real values in its domain?
 6.2.11: In Exercises 1120, verify the identity. 11. sin t csc t = 1 12. sec...
 6.2.12: In Exercises 1120, verify the identity. 11. sin t csc t = 1 12. sec...
 6.2.13: In Exercises 1120, verify the identity. 11. sin t csc t = 1 12. sec...
 6.2.14: In Exercises 1120, verify the identity. 11. sin t csc t = 1 12. sec...
 6.2.15: In Exercises 1120, verify the identity. 11. sin t csc t = 1 12. sec...
 6.2.16: In Exercises 1120, verify the identity. 11. sin t csc t = 1 12. sec...
 6.2.17: In Exercises 1120, verify the identity. 11. sin t csc t = 1 12. sec...
 6.2.18: In Exercises 1120, verify the identity. 11. sin t csc t = 1 12. sec...
 6.2.19: In Exercises 1120, verify the identity. 11. sin t csc t = 1 12. sec...
 6.2.20: In Exercises 1120, verify the identity. 11. sin t csc t = 1 12. sec...
 6.2.21: In Exercises 2130, use a graphing utility to complete the table and...
 6.2.22: In Exercises 2130, use a graphing utility to complete the table and...
 6.2.23: In Exercises 2130, use a graphing utility to complete the table and...
 6.2.24: In Exercises 2130, use a graphing utility to complete the table and...
 6.2.25: In Exercises 2130, use a graphing utility to complete the table and...
 6.2.26: In Exercises 2130, use a graphing utility to complete the table and...
 6.2.27: In Exercises 2130, use a graphing utility to complete the table and...
 6.2.28: In Exercises 2130, use a graphing utility to complete the table and...
 6.2.29: In Exercises 2130, use a graphing utility to complete the table and...
 6.2.30: In Exercises 2130, use a graphing utility to complete the table and...
 6.2.31: In Exercises 31 and 32, describe the error. 31. (1 + tan x)[1 + cot...
 6.2.32: In Exercises 31 and 32, describe the error. 31. (1 + tan x)[1 + cot...
 6.2.33: In Exercises 33 and 34, fill in the missing step(s). 33. sec4 x 2 s...
 6.2.34: In Exercises 33 and 34, fill in the missing step(s). 33. sec4 x 2 s...
 6.2.35: In Exercises 3540, verify the identity. 35. cot( 2 x)csc x = sec x ...
 6.2.36: In Exercises 3540, verify the identity. 35. cot( 2 x)csc x = sec x ...
 6.2.37: In Exercises 3540, verify the identity. 35. cot( 2 x)csc x = sec x ...
 6.2.38: In Exercises 3540, verify the identity. 35. cot( 2 x)csc x = sec x ...
 6.2.39: In Exercises 3540, verify the identity. 35. cot( 2 x)csc x = sec x ...
 6.2.40: In Exercises 3540, verify the identity. 35. cot( 2 x)csc x = sec x ...
 6.2.41: In Exercises 4148, verify the identity algebraically. Use the table...
 6.2.42: In Exercises 4148, verify the identity algebraically. Use the table...
 6.2.43: In Exercises 4148, verify the identity algebraically. Use the table...
 6.2.44: In Exercises 4148, verify the identity algebraically. Use the table...
 6.2.45: In Exercises 4148, verify the identity algebraically. Use the table...
 6.2.46: In Exercises 4148, verify the identity algebraically. Use the table...
 6.2.47: In Exercises 4148, verify the identity algebraically. Use the table...
 6.2.48: In Exercises 4148, verify the identity algebraically. Use the table...
 6.2.49: In Exercises 4960, verify the identity algebraically. Use a graphin...
 6.2.50: In Exercises 4960, verify the identity algebraically. Use a graphin...
 6.2.51: In Exercises 4960, verify the identity algebraically. Use a graphin...
 6.2.52: In Exercises 4960, verify the identity algebraically. Use a graphin...
 6.2.53: In Exercises 4960, verify the identity algebraically. Use a graphin...
 6.2.54: In Exercises 4960, verify the identity algebraically. Use a graphin...
 6.2.55: In Exercises 4960, verify the identity algebraically. Use a graphin...
 6.2.56: In Exercises 4960, verify the identity algebraically. Use a graphin...
 6.2.57: In Exercises 4960, verify the identity algebraically. Use a graphin...
 6.2.58: In Exercises 4960, verify the identity algebraically. Use a graphin...
 6.2.59: In Exercises 4960, verify the identity algebraically. Use a graphin...
 6.2.60: In Exercises 4960, verify the identity algebraically. Use a graphin...
 6.2.61: In Exercises 6164, use a graphing utility to graph the trigonometri...
 6.2.62: In Exercises 6164, use a graphing utility to graph the trigonometri...
 6.2.63: In Exercises 6164, use a graphing utility to graph the trigonometri...
 6.2.64: In Exercises 6164, use a graphing utility to graph the trigonometri...
 6.2.65: In Exercises 65 and 66, use the properties of logarithms and trigon...
 6.2.66: In Exercises 65 and 66, use the properties of logarithms and trigon...
 6.2.67: In Exercises 6770, use the cofunction identities to evaluate the ex...
 6.2.68: In Exercises 6770, use the cofunction identities to evaluate the ex...
 6.2.69: In Exercises 6770, use the cofunction identities to evaluate the ex...
 6.2.70: In Exercises 6770, use the cofunction identities to evaluate the ex...
 6.2.71: In Exercises 7174, powers of trigonometric functions are rewritten ...
 6.2.72: In Exercises 7174, powers of trigonometric functions are rewritten ...
 6.2.73: In Exercises 7174, powers of trigonometric functions are rewritten ...
 6.2.74: In Exercises 7174, powers of trigonometric functions are rewritten ...
 6.2.75: In Exercises 7578, verify the identity. 75. tan(sin1 x) = x 1 x2 76...
 6.2.76: In Exercises 7578, verify the identity. 75. tan(sin1 x) = x 1 x2 76...
 6.2.77: In Exercises 7578, verify the identity. 75. tan(sin1 x) = x 1 x2 76...
 6.2.78: In Exercises 7578, verify the identity. 75. tan(sin1 x) = x 1 x2 76...
 6.2.79: The length s of a shadow cast by a vertical gnomon (a device used t...
 6.2.80: The rate of change of the function f(x) = sin x + csc x is given by...
 6.2.81: In Exercises 81 and 82, determine whether the statement is true or ...
 6.2.82: In Exercises 81 and 82, determine whether the statement is true or ...
 6.2.83: In Exercises 83 and 84, (a) verify the identity and (b) determine w...
 6.2.84: In Exercises 83 and 84, (a) verify the identity and (b) determine w...
 6.2.85: In Exercises 85 88, use the trigonometric substitution to write the...
 6.2.86: In Exercises 85 88, use the trigonometric substitution to write the...
 6.2.87: In Exercises 85 88, use the trigonometric substitution to write the...
 6.2.88: In Exercises 85 88, use the trigonometric substitution to write the...
 6.2.89: In Exercises 89 92, explain why the equation is not an identity and...
 6.2.90: In Exercises 89 92, explain why the equation is not an identity and...
 6.2.91: In Exercises 89 92, explain why the equation is not an identity and...
 6.2.92: In Exercises 89 92, explain why the equation is not an identity and...
 6.2.93: Verify that for all integers n, sin[ (12n + 1) 6 ] = 1 2
 6.2.94: Explain how to use the figure to derive the identity sec2 1 sec2 = ...
 6.2.95: In Exercises 9598, use a graphing utility to construct a table of v...
 6.2.96: In Exercises 9598, use a graphing utility to construct a table of v...
 6.2.97: In Exercises 9598, use a graphing utility to construct a table of v...
 6.2.98: In Exercises 9598, use a graphing utility to construct a table of v...
Solutions for Chapter 6.2: Analytic Trigonometry
Full solutions for Algebra and Trigonometry: Real Mathematics, Real People  7th Edition
ISBN: 9781305071735
Solutions for Chapter 6.2: Analytic Trigonometry
Get Full SolutionsChapter 6.2: Analytic Trigonometry includes 98 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 98 problems in chapter 6.2: Analytic Trigonometry have been answered, more than 59278 students have viewed full stepbystep solutions from this chapter.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

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.)

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

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.

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

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.

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

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

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

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

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

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.

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 .

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).

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

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.

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.

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