 7.1.7.1.1: Each problem that follows refers to triangle ABC.IfA 40, B = 60, an...
 7.1.7.1.2: Each problem that follows refers to triangle ABC.IfA 80, B = 30, an...
 7.1.7.1.3: Each problem that follows refers to triangle ABC.If B 120, C = 200 ...
 7.1.7.1.4: Each problem that follows refers to triangle ABC.IfB = 110, C = 40,...
 7.1.7.1.5: Each problem that follows refers to triangle ABC.IfA 10, C = 100, a...
 7.1.7.1.6: Each problem that follows refers to triangle ABC.IfA 5,C= 125, and ...
 7.1.7.1.7: Each problem that follows refers to triangle ABC.IfA = 50, B = 60, ...
 7.1.7.1.8: Each problem that follows refers to triangle ABC.IfB = 40, C = 70, ...
 7.1.7.1.9: Each problem that follows refers to triangle ABC.IfA = 52, B = 48, ...
 7.1.7.1.10: Each problem that follows refers to triangle ABC.IfA 33, C = 82, an...
 7.1.7.1.11: The following information refers to triangle ABC. In each case, fin...
 7.1.7.1.12: The following information refers to triangle ABC. In each case, fin...
 7.1.7.1.13: The following information refers to triangle ABC. In each case, fin...
 7.1.7.1.14: The following information refers to triangle ABC. In each case, fin...
 7.1.7.1.15: The following information refers to triangle ABC. In each case, fin...
 7.1.7.1.16: The following information refers to triangle ABC. In each case, fin...
 7.1.7.1.17: The following information refers to triangle ABC. In each case, fin...
 7.1.7.1.18: The following information refers to triangle ABC. In each case, fin...
 7.1.7.1.19: Geometry The circle in Figure 12 has a radius of r and center at C....
 7.1.7.1.20: Geometry The circle in Figure 12 has a radius of r and center at C....
 7.1.7.1.21: Geometry The circle in Figure 12 has a radius of r and center at C....
 7.1.7.1.22: Geometry The circle in Figure 12 has a radius of r and center at C....
 7.1.7.1.23: Geometry The circle in Figure 12 has a radius of r and center at C....
 7.1.7.1.24: Geometry The circle in Figure 12 has a radius of r and center at C....
 7.1.7.1.25: Angle of Elevation A man standing near a radio station antenna obse...
 7.1.7.1.26: Angle of Elevation A person standing on the street looks up to the ...
 7.1.7.1.27: Angle of Depression A man is flying in a hotair balloon in a strai...
 7.1.7.1.28: Angle of Elevation A woman entering an outside glass elevator on th...
 7.1.7.1.29: Angle of Elevation From a point on the ground, a person notices tha...
 7.1.7.1.30: Angle of Elevation A 155foot antenna is on top of a tall building....
 7.1.7.1.31: Height of a Tree Figure 16 is a diagram that shows how Colleen esti...
 7.1.7.1.32: Sea Rescue A helicopter makes a forced landing at sea. The last rad...
 7.1.7.1.33: Distance to a Ship A ship is anchored off a long straight shoreline...
 7.1.7.1.34: Distance to a Rocket Tom and Fred are 3.5 miles apart watching a ro...
 7.1.7.1.35: Force A tightrope walker is standing still with one foot on the tig...
 7.1.7.1.36: Force A tightrope walker weighing 145 pounds is standing still at t...
 7.1.7.1.37: Force If you have ever ridden on a chair lift at a ski area and had...
 7.1.7.1.38: Force A chair lift at a ski resort is stopped halfway between two p...
 7.1.7.1.39: The problems that follow review material we covered in Sections 3.1...
 7.1.7.1.40: The problems that follow review material we covered in Sections 3.1...
 7.1.7.1.41: The problems that follow review material we covered in Sections 3.1...
 7.1.7.1.42: The problems that follow review material we covered in Sections 3.1...
 7.1.7.1.43: The problems that follow review material we covered in Sections 3.1...
 7.1.7.1.44: The problems that follow review material we covered in Sections 3.1...
 7.1.7.1.45: Find eto the nearest tenth of a degree if 0 s: e< 3600 , and sin e ...
 7.1.7.1.46: Find eto the nearest tenth of a degree if 0 s: e< 3600 , and sin e ...
Solutions for Chapter 7.1: Triangles
Full solutions for Trigonometry
ISBN: 9780495108351
Solutions for Chapter 7.1: Triangles
Get Full SolutionsTrigonometry was written by and is associated to the ISBN: 9780495108351. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 7.1: Triangles includes 46 full stepbystep solutions. This textbook survival guide was created for the textbook: Trigonometry, edition: . Since 46 problems in chapter 7.1: Triangles have been answered, more than 29888 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.

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).

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

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

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 .

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

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).

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.

Outer product uv T
= column times row = rank one matrix.

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

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

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(DÂ» O.

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

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

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

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

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