 7.215: 1 through 10 refer to triangle ABC, which is not necessarily a righ...
 7.216: 1 through 10 refer to triangle ABC, which is not necessarily a righ...
 7.217: 1 through 10 refer to triangle ABC, which is not necessarily a righ...
 7.218: 1 through 10 refer to triangle ABC, which is not necessarily a righ...
 7.219: 1 through 10 refer to triangle ABC, which is not necessarily a righ...
 7.220: 1 through 10 refer to triangle ABC, which is not necessarily a righ...
 7.221: 1 through 10 refer to triangle ABC, which is not necessarily a righ...
 7.222: 1 through 10 refer to triangle ABC, which is not necessarily a righ...
 7.223: 1 through 10 refer to triangle ABC, which is not necessarily a righ...
 7.224: 1 through 10 refer to triangle ABC, which is not necessarily a righ...
 7.225: Find the area of the triangle in 2.
 7.226: Find the area of the triangle in 2.
 7.227: Find the area of the triangle in 8.
 7.228: Geometry The two equal sides of an isosceles triangle are each 38 c...
 7.229: Angle of Elevation A man standing near a building notices that the ...
 7.230: Geometry The diagonals of a parallelogram are 26.8 meters and 39.4 ...
 7.231: Arc Length Suppose Figure 1 is an exaggerated diagram of a plane fi...
 7.232: Distance and Bearing A man wandering in the desert walks 3.3 miles ...
 7.233: Distance Two guy wires from the top of a tent pole are anchored to ...
 7.234: Ground Speed A plane is headed due east with an airspeed of 345 mil...
 7.235: Height of a Tree To estimate the height of a tree, two people posit...
 7.236: True Course and Speed A plane fiying with an airspeed of 325 miles ...
 7.237: Let V = 5i + 12j, V = 4i + j, and W = i  4j, and find lvl
 7.238: Let V = 5i + 12j, V = 4i + j, and W = i  4j, and find 3V + 5V
 7.239: Let V = 5i + 12j, V = 4i + j, and W = i  4j, and find 12v  wi
 7.240: Let V = 5i + 12j, V = 4i + j, and W = i  4j, and find VW
 7.241: The angle between V and V to the nearest tenth of a degree.
 7.242: Show that V = 3i + 6j and W =  8i + 4j are perpendicular.
 7.243: Find the value of b so that vectors U = 5i + l2j and V = 4i + bj ar...
 7.244: Find the work performed by the force F 33i  4j in moving an object...
 7.245: Use the fact that Earth completes one revolution about the Sun ever...
 7.246: Find the distanec AB using triangle ABC.
 7.247: Because triangle ABC is isosceles, angles CAB and CBA are equal. Fi...
 7.248: Use triangle ABD to find the length of AD.
 7.249: Find the length of CD. Give your answer both in terms of AU and in ...
Solutions for Chapter 7: Triangles
Full solutions for Trigonometry
ISBN: 9780495108351
Solutions for Chapter 7: Triangles
Get Full SolutionsSince 35 problems in chapter 7: Triangles have been answered, more than 31316 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Trigonometry, edition: . Trigonometry was written by and is associated to the ISBN: 9780495108351. Chapter 7: Triangles includes 35 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

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.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

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.

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.

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.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

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.

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

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.

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(D» O.

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).

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

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.