 1.4.1: Given cos u = 1sec u , two equivalent forms of this identity are se...
 1.4.2: Given tan u = 1cot u , two equivalent forms of this identity are co...
 1.4.3: For an angle u measuring 105, the trigonometric functions andare po...
 1.4.4: If sin u 7 0 and tan u 7 0, then u is in quadrant .
 1.4.5: Determine whether each statement is possible or impossible sin u = ...
 1.4.6: Determine whether each statement is possible or impossible tan u = ...
 1.4.7: Determine whether each statement is possible or impossible sin u 7 ...
 1.4.8: Determine whether each statement is possible or impossible cos u = 1.5
 1.4.9: Determine whether each statement is possible or impossible cot u = ...
 1.4.10: Determine whether each statement is possible or impossible sin2 u +...
 1.4.11: Use the appropriate reciprocal identity to find each function value...
 1.4.12: Use the appropriate reciprocal identity to find each function value...
 1.4.13: Use the appropriate reciprocal identity to find each function value...
 1.4.14: Use the appropriate reciprocal identity to find each function value...
 1.4.15: Use the appropriate reciprocal identity to find each function value...
 1.4.16: Use the appropriate reciprocal identity to find each function value...
 1.4.17: Use the appropriate reciprocal identity to find each function value...
 1.4.18: Use the appropriate reciprocal identity to find each function value...
 1.4.19: Use the appropriate reciprocal identity to find each function value...
 1.4.20: Use the appropriate reciprocal identity to find each function value...
 1.4.21: Use the appropriate reciprocal identity to find each function value...
 1.4.22: Use the appropriate reciprocal identity to find each function value...
 1.4.23: Use the appropriate reciprocal identity to find each function value...
 1.4.24: Use the appropriate reciprocal identity to find each function value...
 1.4.25: Concept Check What is wrong with the following item that appears on...
 1.4.26: Concept Check What is wrong with the statement tan 90 = 1 cot 90 ?
 1.4.27: Determine the signs of the trigonometric functions of an angle in s...
 1.4.28: Determine the signs of the trigonometric functions of an angle in s...
 1.4.29: Determine the signs of the trigonometric functions of an angle in s...
 1.4.30: Determine the signs of the trigonometric functions of an angle in s...
 1.4.31: Determine the signs of the trigonometric functions of an angle in s...
 1.4.32: Determine the signs of the trigonometric functions of an angle in s...
 1.4.33: Determine the signs of the trigonometric functions of an angle in s...
 1.4.34: Determine the signs of the trigonometric functions of an angle in s...
 1.4.35: Determine the signs of the trigonometric functions of an angle in s...
 1.4.36: Determine the signs of the trigonometric functions of an angle in s...
 1.4.37: Determine the signs of the trigonometric functions of an angle in s...
 1.4.38: Determine the signs of the trigonometric functions of an angle in s...
 1.4.39: dentify the quadrant (or possible quadrants) of an angle u that sat...
 1.4.40: dentify the quadrant (or possible quadrants) of an angle u that sat...
 1.4.41: dentify the quadrant (or possible quadrants) of an angle u that sat...
 1.4.42: dentify the quadrant (or possible quadrants) of an angle u that sat...
 1.4.43: dentify the quadrant (or possible quadrants) of an angle u that sat...
 1.4.44: dentify the quadrant (or possible quadrants) of an angle u that sat...
 1.4.45: dentify the quadrant (or possible quadrants) of an angle u that sat...
 1.4.46: dentify the quadrant (or possible quadrants) of an angle u that sat...
 1.4.47: dentify the quadrant (or possible quadrants) of an angle u that sat...
 1.4.48: dentify the quadrant (or possible quadrants) of an angle u that sat...
 1.4.49: dentify the quadrant (or possible quadrants) of an angle u that sat...
 1.4.50: dentify the quadrant (or possible quadrants) of an angle u that sat...
 1.4.51: Why are the answers to Exercises 41 and 45 the same?
 1.4.52: Why is there no angle u that satisfies tan u 7 0, cot u 6 0?
 1.4.53: Determine whether each statement is possible or impossible. See Exa...
 1.4.54: Determine whether each statement is possible or impossible. See Exa...
 1.4.55: Determine whether each statement is possible or impossible. See Exa...
 1.4.56: Determine whether each statement is possible or impossible. See Exa...
 1.4.57: Determine whether each statement is possible or impossible. See Exa...
 1.4.58: Determine whether each statement is possible or impossible. See Exa...
 1.4.59: Determine whether each statement is possible or impossible. See Exa...
 1.4.60: Determine whether each statement is possible or impossible. See Exa...
 1.4.61: Determine whether each statement is possible or impossible. See Exa...
 1.4.62: Determine whether each statement is possible or impossible. See Exa...
 1.4.63: Determine whether each statement is possible or impossible. See Exa...
 1.4.64: Determine whether each statement is possible or impossible. See Exa...
 1.4.65: Use identities to solve each of the following. Rationalize denomina...
 1.4.66: Use identities to solve each of the following. Rationalize denomina...
 1.4.67: Use identities to solve each of the following. Rationalize denomina...
 1.4.68: Use identities to solve each of the following. Rationalize denomina...
 1.4.69: Use identities to solve each of the following. Rationalize denomina...
 1.4.70: Use identities to solve each of the following. Rationalize denomina...
 1.4.71: Use identities to solve each of the following. Rationalize denomina...
 1.4.72: Use identities to solve each of the following. Rationalize denomina...
 1.4.73: Give all six trigonometric function values for each angle u. Ration...
 1.4.74: Give all six trigonometric function values for each angle u. Ration...
 1.4.75: Give all six trigonometric function values for each angle u. Ration...
 1.4.76: Give all six trigonometric function values for each angle u. Ration...
 1.4.77: Give all six trigonometric function values for each angle u. Ration...
 1.4.78: Give all six trigonometric function values for each angle u. Ration...
 1.4.79: Give all six trigonometric function values for each angle u. Ration...
 1.4.80: Give all six trigonometric function values for each angle u. Ration...
 1.4.81: Give all six trigonometric function values for each angle u. Ration...
 1.4.82: Give all six trigonometric function values for each angle u. Ration...
 1.4.83: Give all six trigonometric function values for each angle u. Ration...
 1.4.84: Give all six trigonometric function values for each angle u. Ration...
 1.4.85: Work each problem Derive the identity 1 + cot2 u = csc2 u by dividi...
 1.4.86: Work each problem Derive the quotient identity cos usin u = cot u.
 1.4.87: Work each problem Concept Check True or false: For all angles u, si...
 1.4.88: Work each problem Concept Check True or false: Since cot u = cos u ...
 1.4.89: Concept Check Suppose that 90 6 u 6 180. Find the sign of each func...
 1.4.90: Concept Check Suppose that 90 6 u 6 180. Find the sign of each func...
 1.4.91: Concept Check Suppose that 90 6 u 6 180. Find the sign of each func...
 1.4.92: Concept Check Suppose that 90 6 u 6 180. Find the sign of each func...
 1.4.93: Concept Check Suppose that 90 6 u 6 180. Find the sign of each func...
 1.4.94: Concept Check Suppose that 90 6 u 6 180. Find the sign of each func...
 1.4.95: Concept Check Suppose that 90 6 u 6 180. Find the sign of each func...
 1.4.96: Concept Check Suppose that 90 6 u 6 180. Find the sign of each func...
 1.4.97: Concept Check Suppose that 90 6 u 6 90. Find the sign of each func...
 1.4.98: Concept Check Suppose that 90 6 u 6 90. Find the sign of each func...
 1.4.99: Concept Check Suppose that 90 6 u 6 90. Find the sign of each func...
 1.4.100: Concept Check Suppose that 90 6 u 6 90. Find the sign of each func...
 1.4.101: Concept Check Suppose that 90 6 u 6 90. Find the sign of each func...
 1.4.102: Concept Check Suppose that 90 6 u 6 90. Find the sign of each func...
 1.4.103: Concept Check Suppose that 90 6 u 6 90. Find the sign of each func...
 1.4.104: Concept Check Suppose that 90 6 u 6 90. Find the sign of each func...
 1.4.105: Concept Check Find a solution for each equation. tan13u  42 = 1cot...
 1.4.106: Concept Check Find a solution for each equation. cos16u + 52 = 1sec...
 1.4.107: Concept Check Find a solution for each equation. sin14u + 22 csc13u...
 1.4.108: Concept Check Find a solution for each equation. sec12u + 62 cos15u...
 1.4.109: Concept Check The screen below was obtained with the calculator in ...
 1.4.110: Concept Check The screen below was obtained with the calculator in ...
Solutions for Chapter 1.4: Using the Definitions of the Trigonometric Functions
Full solutions for Trigonometry  11th Edition
ISBN: 9780134217437
Solutions for Chapter 1.4: Using the Definitions of the Trigonometric Functions
Get Full SolutionsSince 110 problems in chapter 1.4: Using the Definitions of the Trigonometric Functions have been answered, more than 23229 students have viewed full stepbystep solutions from this chapter. Trigonometry was written by and is associated to the ISBN: 9780134217437. Chapter 1.4: Using the Definitions of the Trigonometric Functions includes 110 full stepbystep solutions. This textbook survival guide was created for the textbook: Trigonometry, edition: 11. This expansive textbook survival guide covers the following chapters and their solutions.

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.

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

Column space C (A) =
space of all combinations of the columns of A.

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

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.

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and 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.

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 .

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

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.

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

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.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

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!

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.