 6.2.6.1.81: Solve eaeh equation for (j if 0 s (j < 360. V3 sec (j = 2
 6.2.6.1.82: Solve eaeh equation for (j if 0 s (j < 360. v2 esc (j = 2
 6.2.6.1.83: Solve eaeh equation for (j if 0 s (j < 360. v2 esc (j + 5 3
 6.2.6.1.84: Solve eaeh equation for (j if 0 s (j < 360. 2V3 sec (j + 7 3
 6.2.6.1.85: Solve eaeh equation for (j if 0 s (j < 360. 4 sin (j 2 esc (j 0
 6.2.6.1.86: Solve eaeh equation for (j if 0 s (j < 360. 4 cos (j 3 sec (j = 0
 6.2.6.1.87: Solve eaeh equation for (j if 0 s (j < 360. sec (j  2 tan (j = 0
 6.2.6.1.88: Solve eaeh equation for (j if 0 s (j < 360. esc (j + 2 cot (j 0
 6.2.6.1.89: Solve eaeh equation for (j if 0 s (j < 360. sin 2(j  cos (j = 0
 6.2.6.1.90: Solve eaeh equation for (j if 0 s (j < 360. 2 sin (j + sin 2(j = 0
 6.2.6.1.91: Solve eaeh equation for (j if 0 s (j < 360. 2 cos (j + 1 sec
 6.2.6.1.92: Solve eaeh equation for (j if 0 s (j < 360. 2 sin (j  1 esc (j
 6.2.6.1.93: Solve each equation for x if 0 s x < 21T. Give your answers in radi...
 6.2.6.1.94: Solve each equation for x if 0 s x < 21T. Give your answers in radi...
 6.2.6.1.95: Solve each equation for x if 0 s x < 21T. Give your answers in radi...
 6.2.6.1.96: Solve each equation for x if 0 s x < 21T. Give your answers in radi...
 6.2.6.1.97: Solve each equation for x if 0 s x < 21T. Give your answers in radi...
 6.2.6.1.98: Solve each equation for x if 0 s x < 21T. Give your answers in radi...
 6.2.6.1.99: Solve each equation for x if 0 s x < 21T. Give your answers in radi...
 6.2.6.1.100: Solve each equation for x if 0 s x < 21T. Give your answers in radi...
 6.2.6.1.101: Solve each equation for x if 0 s x < 21T. Give your answers in radi...
 6.2.6.1.102: Solve each equation for x if 0 s x < 21T. Give your answers in radi...
 6.2.6.1.103: Solve each equation for x if 0 s x < 21T. Give your answers in radi...
 6.2.6.1.104: Solve each equation for x if 0 s x < 21T. Give your answers in radi...
 6.2.6.1.105: Solve for (j if 0 :::; (j < 3600 V3 sin (j + cos (j = V3
 6.2.6.1.106: Solve for (j if 0 :::; (j < 3600 sin e V3 cos (j V3
 6.2.6.1.107: Solve for (j if 0 :::; (j < 3600 V3 sin (j  cos (j = I
 6.2.6.1.108: Solve for (j if 0 :::; (j < 3600 sin (j  cos (j I
 6.2.6.1.109: Solve for (j if 0 :::; (j < 3600 sin  cos (j 0
 6.2.6.1.110: Solve for (j if 0 :::; (j < 3600 sin 2 + cos (j
 6.2.6.1.111: Solve for (j if 0 :::; (j < 3600 cos "2 cos (j = 1
 6.2.6.1.112: Solve for (j if 0 :::; (j < 3600 cos"2 cos (j 0
 6.2.6.1.113: For each equation, find all degree solutions in the interval 0 s (j...
 6.2.6.1.114: For each equation, find all degree solutions in the interval 0 s (j...
 6.2.6.1.115: For each equation, find all degree solutions in the interval 0 s (j...
 6.2.6.1.116: For each equation, find all degree solutions in the interval 0 s (j...
 6.2.6.1.117: For each equation, find all degree solutions in the interval 0 s (j...
 6.2.6.1.118: For each equation, find all degree solutions in the interval 0 s (j...
 6.2.6.1.119: Write expressions that give all solutions to the equations you solv...
 6.2.6.1.120: Write expressions that give all solutions to the equations you solv...
 6.2.6.1.121: Write expressions that give all solutions to the equations you solv...
 6.2.6.1.122: Write expressions that give all solutions to the equations you solv...
 6.2.6.1.123: Write expressions that give all solutions to the equations you solv...
 6.2.6.1.124: Write expressions that give all solutions to the equations you solv...
 6.2.6.1.125: Physiology In the human body, the value of ethat makes the followin...
 6.2.6.1.126: Physiology Find the value of f) that makes the expression in zero, ...
 6.2.6.1.127: Solving the following equations will require you to use the quadrat...
 6.2.6.1.128: Solving the following equations will require you to use the quadrat...
 6.2.6.1.129: Solving the following equations will require you to use the quadrat...
 6.2.6.1.130: Solving the following equations will require you to use the quadrat...
 6.2.6.1.131: Solving the following equations will require you to use the quadrat...
 6.2.6.1.132: Solving the following equations will require you to use the quadrat...
 6.2.6.1.133: Use your graphing calculator to find all radian solutions in the in...
 6.2.6.1.134: Use your graphing calculator to find all radian solutions in the in...
 6.2.6.1.135: Use your graphing calculator to find all radian solutions in the in...
 6.2.6.1.136: Use your graphing calculator to find all radian solutions in the in...
 6.2.6.1.137: Use your graphing calculator to find all radian solutions in the in...
 6.2.6.1.138: Use your graphing calculator to find all radian solutions in the in...
 6.2.6.1.139: The problems that follow review material we covered in Section 5.4....
 6.2.6.1.140: The problems that follow review material we covered in Section 5.4....
 6.2.6.1.141: The problems that follow review material we covered in Section 5.4....
 6.2.6.1.142: The problems that follow review material we covered in Section 5.4....
 6.2.6.1.143: The problems that follow review material we covered in Section 5.4....
 6.2.6.1.144: The problems that follow review material we covered in Section 5.4....
 6.2.6.1.145: Use a halfangle formula to find sin 22.5.
 6.2.6.1.146: Use a halfangle formula to find cos 15.
Solutions for Chapter 6.2: Equations
Full solutions for Trigonometry
ISBN: 9780495108351
Solutions for Chapter 6.2: Equations
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Since 66 problems in chapter 6.2: Equations have been answered, more than 32200 students have viewed full stepbystep solutions from this chapter. Chapter 6.2: Equations includes 66 full stepbystep solutions. Trigonometry was written by and is associated to the ISBN: 9780495108351. This textbook survival guide was created for the textbook: Trigonometry, edition: .

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

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.

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

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

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.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

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.

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.

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.

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

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.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.