 6.3.1: Suppose you are solving a trigonometric equation for solutions over...
 6.3.2: Suppose you are solving a trigonometric equation for solutions over...
 6.3.3: Suppose you are solving a trigonometric equation for solutions over...
 6.3.4: Suppose you are solving a trigonometric equation for solutions over...
 6.3.5: Explain what is WRONG with the following solution. Solve tan 2u = 2...
 6.3.6: The equation cot x2  csc x2  1 = 0 has no solution over the inter...
 6.3.7: Write each expression as a product of trigonometric functions. See ...
 6.3.8: Write each expression as a product of trigonometric functions. See ...
 6.3.9: Write each expression as a product of trigonometric functions. See ...
 6.3.10: Write each expression as a product of trigonometric functions. See ...
 6.3.11: Write each expression as a product of trigonometric functions. See ...
 6.3.12: Write each expression as a product of trigonometric functions. See ...
 6.3.13: Write each expression as a product of trigonometric functions. See ...
 6.3.14: Write each expression as a product of trigonometric functions. See ...
 6.3.15: Write each expression as a product of trigonometric functions. See ...
 6.3.16: Write each expression as a product of trigonometric functions. See ...
 6.3.17: Write each expression as a product of trigonometric functions. See ...
 6.3.18: Write each expression as a product of trigonometric functions. See ...
 6.3.19: Write each expression as a product of trigonometric functions. See ...
 6.3.20: Write each expression as a product of trigonometric functions. See ...
 6.3.21: Write each expression as a product of trigonometric functions. See ...
 6.3.22: Write each expression as a product of trigonometric functions. See ...
 6.3.23: Write each expression as a product of trigonometric functions. See ...
 6.3.24: Write each expression as a product of trigonometric functions. See ...
 6.3.25: Solve each equation (x in radians and u in degrees) for all exact s...
 6.3.26: Solve each equation (x in radians and u in degrees) for all exact s...
 6.3.27: Solve each equation (x in radians and u in degrees) for all exact s...
 6.3.28: Solve each equation (x in radians and u in degrees) for all exact s...
 6.3.29: Solve each equation (x in radians and u in degrees) for all exact s...
 6.3.30: Solve each equation (x in radians and u in degrees) for all exact s...
 6.3.31: Solve each equation (x in radians and u in degrees) for all exact s...
 6.3.32: Solve each equation (x in radians and u in degrees) for all exact s...
 6.3.33: Solve each equation (x in radians and u in degrees) for all exact s...
 6.3.34: Solve each equation (x in radians and u in degrees) for all exact s...
 6.3.35: Solve each equation (x in radians and u in degrees) for all exact s...
 6.3.36: Solve each equation (x in radians and u in degrees) for all exact s...
 6.3.37: Solve each equation (x in radians and u in degrees) for all exact s...
 6.3.38: Solve each equation (x in radians and u in degrees) for all exact s...
 6.3.39: Solve each equation (x in radians and u in degrees) for all exact s...
 6.3.40: Solve each equation (x in radians and u in degrees) for all exact s...
 6.3.41: Solve each equation for solutions over the interval 30, 2p2. Write ...
 6.3.42: Solve each equation for solutions over the interval 30, 2p2. Write ...
 6.3.43: Solve each equation for solutions over the interval 30, 2p2. Write ...
 6.3.44: Solve each equation for solutions over the interval 30, 2p2. Write ...
 6.3.45: The following equations cannot be solved by algebraic methods. Use ...
 6.3.46: The following equations cannot be solved by algebraic methods. Use ...
 6.3.47: Pressure of a Plucked String If a string with a fundamental frequen...
 6.3.48: Hearing Beats in Music Musicians sometimes tune instruments by play...
 6.3.49: Hearing Difference Tones When a musical instrument creates a tone o...
 6.3.50: Daylight Hours in New Orleans The seasonal variation in length of d...
 6.3.51: Average Monthly Temperature in Vancouver The following function app...
 6.3.52: Average Monthly Temperature in Phoenix The following function appro...
 6.3.53: for time t in seconds, where i is instantaneous current in amperes,...
 6.3.54: for time t in seconds, where i is instantaneous current in amperes,...
 6.3.55: for time t in seconds, where i is instantaneous current in amperes,...
 6.3.56: for time t in seconds, where i is instantaneous current in amperes,...
Solutions for Chapter 6.3: Trigonometric Equations II
Full solutions for Trigonometry  10th Edition
ISBN: 9780321671776
Solutions for Chapter 6.3: Trigonometric Equations II
Get Full SolutionsChapter 6.3: Trigonometric Equations II includes 56 full stepbystep solutions. This textbook survival guide was created for the textbook: Trigonometry, edition: 10. This expansive textbook survival guide covers the following chapters and their solutions. Trigonometry was written by and is associated to the ISBN: 9780321671776. Since 56 problems in chapter 6.3: Trigonometric Equations II have been answered, more than 35592 students have viewed full stepbystep solutions from this chapter.

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.

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.

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).

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

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

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

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.

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 .

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

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.

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

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

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·

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

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