 7.2.7.1.47: For each of the following triangles, solve for B and use the result...
 7.2.7.1.48: For each of the following triangles, solve for B and use the result...
 7.2.7.1.49: For each of the following triangles, solve for B and use the result...
 7.2.7.1.50: For each of the following triangles, solve for B and use the result...
 7.2.7.1.51: For each of the following triangles, solve for B and use the result...
 7.2.7.1.52: For each of the following triangles, solve for B and use the result...
 7.2.7.1.53: Find all solutions to each of the following triangles:A = 38, a = 4...
 7.2.7.1.54: Find all solutions to each of the following triangles:A = 43, a = 3...
 7.2.7.1.55: Find all solutions to each of the following triangles:A = 112.2, a ...
 7.2.7.1.56: Find all solutions to each of the following triangles:A = 124.3, a ...
 7.2.7.1.57: Find all solutions to each of the following triangles:C = 27 50', C...
 7.2.7.1.58: Find all solutions to each of the following triangles:C=5130',c=707...
 7.2.7.1.59: Find all solutions to each of the following triangles:B = 45 10', b...
 7.2.7.1.60: Find all solutions to each of the following triangles:B = 62 40', b...
 7.2.7.1.61: Find all solutions to each of the following triangles:B = 118, b = ...
 7.2.7.1.62: Find all solutions to each of the following triangles:B = 30, b = 4...
 7.2.7.1.63: Find all solutions to each of the following triangles:A = 142, b = ...
 7.2.7.1.64: Find all solutions to each of the following triangles:A = 65, b = 7...
 7.2.7.1.65: Find all solutions to each of the following triangles:C = 26.8, c =...
 7.2.7.1.66: Find all solutions to each of the following triangles:C = 73.4, c =...
 7.2.7.1.67: Distance A 51foot wire running from the top of a tent pole to the ...
 7.2.7.1.68: Distance A hotair balloon is held at a constant altitude by two ro...
 7.2.7.1.69: Draw vectors representing the course of a ship that travels 75 mile...
 7.2.7.1.70: Draw vectors representing the course of a ship that travels 75 mile...
 7.2.7.1.71: Draw vectors representing the course of a ship that travels 25 mile...
 7.2.7.1.72: Draw vectors representing the course of a ship that travels 25 mile...
 7.2.7.1.73: Ground Speed A plane is headed due east with an airspeed of 340 mil...
 7.2.7.1.74: Current A ship is headed due north at a constant 16 miles per hour....
 7.2.7.1.75: Ground Speed A ship headed due east is moving through the water at ...
 7.2.7.1.76: True Course A plane headed due east is traveling with an airspeed o...
 7.2.7.1.77: Leaniug Wiudmill After a wind storm, a farmer notices that his 32f...
 7.2.7.1.78: Distance A boy is riding his motorcycle on a road that runs east an...
 7.2.7.1.79: The problems that follow review material we covered in Section 6.2....
 7.2.7.1.80: The problems that follow review material we covered in Section 6.2....
 7.2.7.1.81: The problems that follow review material we covered in Section 6.2....
 7.2.7.1.82: The problems that follow review material we covered in Section 6.2....
 7.2.7.1.83: The problems that follow review material we covered in Section 6.2....
 7.2.7.1.84: The problems that follow review material we covered in Section 6.2....
 7.2.7.1.85: Find all radian solutions using exact values only.2 cos x sec x + t...
 7.2.7.1.86: Find all radian solutions using exact values only.2 cos2 X  sin x = 1
 7.2.7.1.87: Find all radian solutions using exact values only.sin x + cos x = 0
 7.2.7.1.88: Find all radian solutions using exact values only.sinx cosx 1
Solutions for Chapter 7.2: Triangles
Full solutions for Trigonometry
ISBN: 9780495108351
Solutions for Chapter 7.2: Triangles
Get Full SolutionsTrigonometry was written by and is associated to the ISBN: 9780495108351. Chapter 7.2: Triangles includes 42 full stepbystep solutions. This textbook survival guide was created for the textbook: Trigonometry, edition: . Since 42 problems in chapter 7.2: Triangles have been answered, more than 46623 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions.

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).

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

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

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.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

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 .

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

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

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

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

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).

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

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

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.