 2.4.1: Leading NFL Receiver As of the end of the 2009 National Football Le...
 2.4.2: Height of Mt. Everest When Mt. Everest was first surveyed, the surv...
 2.4.3: Longest Vehicular Tunnel The E. Johnson Memorial Tunnel in Colorado...
 2.4.4: Top WNBA Scorer Womens National Basketball Association player Cappi...
 2.4.5: Circumference of a Circle The formula for the circumference of a ci...
 2.4.6: Explain the distinction between a measurement of 23.0 ft and a meas...
 2.4.7: If h is the actual height of a building and the height is measured ...
 2.4.8: If w is the actual weight of a car and the weight is measured as 15...
 2.4.9: Solve each right triangle. When two sides are given, give angles in...
 2.4.10: Solve each right triangle. When two sides are given, give angles in...
 2.4.11: Solve each right triangle. When two sides are given, give angles in...
 2.4.12: Solve each right triangle. When two sides are given, give angles in...
 2.4.13: Solve each right triangle. When two sides are given, give angles in...
 2.4.14: Solve each right triangle. When two sides are given, give angles in...
 2.4.15: Solve each right triangle. When two sides are given, give angles in...
 2.4.16: Solve each right triangle. When two sides are given, give angles in...
 2.4.17: Can a right triangle be solved if we are given measures of its two ...
 2.4.18: Concept Check If we are given an acute angle and a side in a right ...
 2.4.19: Explain why you can always solve a right triangle if you know the m...
 2.4.20: Explain why you can always solve a right triangle if you know the l...
 2.4.21: Solve each right triangle. In each case, C = 90. If angle informati...
 2.4.22: Solve each right triangle. In each case, C = 90. If angle informati...
 2.4.23: Solve each right triangle. In each case, C = 90. If angle informati...
 2.4.24: Solve each right triangle. In each case, C = 90. If angle informati...
 2.4.25: Solve each right triangle. In each case, C = 90. If angle informati...
 2.4.26: Solve each right triangle. In each case, C = 90. If angle informati...
 2.4.27: Solve each right triangle. In each case, C = 90. If angle informati...
 2.4.28: Solve each right triangle. In each case, C = 90. If angle informati...
 2.4.29: Solve each right triangle. In each case, C = 90. If angle informati...
 2.4.30: Solve each right triangle. In each case, C = 90. If angle informati...
 2.4.31: Solve each right triangle. In each case, C = 90. If angle informati...
 2.4.32: Solve each right triangle. In each case, C = 90. If angle informati...
 2.4.33: Solve each right triangle. In each case, C = 90. If angle informati...
 2.4.34: Solve each right triangle. In each case, C = 90. If angle informati...
 2.4.35: Solve each right triangle. In each case, C = 90. If angle informati...
 2.4.36: Solve each right triangle. In each case, C = 90. If angle informati...
 2.4.37: Explain the meaning of the term angle of elevation.
 2.4.38: Concept Check Can an angle of elevation be more than 90?
 2.4.39: Explain why the angle of depression DAB has the same measure as the...
 2.4.40: Why is angle CAB not an angle of depression in the figure for Exerc...
 2.4.41: Height of a Ladder on a Wall A 13.5m fire truck ladder is leaning ...
 2.4.42: Distance across a Lake To find the distance RS across a lake, a sur...
 2.4.43: Height of a Building From a window 30.0 ft above the street, the an...
 2.4.44: Diameter of the Sun To determine the diameter of the sun, an astron...
 2.4.45: Side Lengths of a Triangle The length of the base of an isosceles t...
 2.4.46: Altitude of a Triangle Find the altitude of an isosceles triangle h...
 2.4.47: Angle of Elevation of the Pyramid of the Sun The Pyramid of the Sun...
 2.4.48: Cloud Ceiling The U.S. Weather Bureau defines a cloud ceiling as th...
 2.4.49: Height of a Tower The shadow of a vertical tower is 40.6 m long whe...
 2.4.50: Distance from the Ground to the Top of a Building The angle of depr...
 2.4.51: Length of a Shadow Suppose that the angle of elevation of the sun i...
 2.4.52: Airplane Distance An airplane is flying 10,500 ft above level groun...
 2.4.53: Angle of Depression of a Light A company safety committee has recom...
 2.4.54: Height of a Building The angle of elevation from the top of a small...
 2.4.55: Angle of Elevation of the Sun The length of the shadow of a buildin...
 2.4.56: Angle of Elevation of the Sun The length of the shadow of a flagpol...
 2.4.57: Height of Mt. Everest The highest mountain peak in the world is Mt....
 2.4.58: Error in Measurement A degree may seem like a very small unit, but ...
Solutions for Chapter 2.4: Solving Right Triangles
Full solutions for Trigonometry  10th Edition
ISBN: 9780321671776
Solutions for Chapter 2.4: Solving Right Triangles
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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!

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

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

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

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

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.

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

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

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

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

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).

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

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