 135.1: B A C 35
 135.2: A C 70 40
 135.3: A 42, b 57, a 63 4
 135.4: a 5, b 12, c 13
 135.5: Find the distance from the pitchers mound to first base.
 135.6: Find the angle between home plate, thepitchers mound, and first base.
 135.7: 72 48 13
 135.8: 15 19 18
 135.9: 185 140 1
 135.10: 34 15 12
 135.11: 17 11
 135.12: 71
 135.13: a 20, c 24, B 47
 135.14: a 345, b 648, c 442
 135.15: A 36, a 10 , b 19
 135.16: A 25, B 78, a 13.7
 135.17: a 21.5, b 16.7, c 10.3
 135.18: a 16, b 24, c 41
 135.19: GEOMETRY In rhombus ABCD, the measure of ADC is 52. Find the measur...
 135.20: SURVEYING Two sides of a triangular plot of land have lengths of 42...
 135.21: a 8, b 24, c 18
 135.22: B 19, a 51, c 61
 135.23: A 56, B 22, a 12.2
 135.24: a 4, b 8, c 5
 135.25: a 21.5, b 13, C 38
 135.26: A 40, b 7, a 6
 135.27: An anthropologist examining the footprints made by a bipedal (twof...
 135.28: Find the step angle made by the hindfeet of a herbivorous dinosaur ...
 135.29: An efficient walker has a step angle that approaches 180, meaning t...
 135.30: AVIATION A pilot typically flies a route from Bloomington to Rockfo...
 135.31: REASONING Explain how to solve a triangle by using the Law of Cosin...
 135.32: FIND THE ERROR Mateo and Amy are deciding which method, the Law of ...
 135.33: OPEN ENDED Give an example of a triangle that can be solved by firs...
 135.34: CHALLENGE Explain how the Pythagorean Theorem is a special case of ...
 135.35: Writing in Math Use the information on page 793 to explain how you ...
 135.36: ACT/SAT In DEF, what is the value of to the nearest degree? A 26 B ...
 135.37: REVIEW Two trucks, A and B, start from the intersection C of two st...
 135.38: SANDBOX Mr. Blackwell is building a triangular sandbox. He is to jo...
 135.39: (5, 12)
 135.40: (4, 7)
 135.41: (10 , 6 )
 135.42: ex + 5 = 9 4
 135.43: 4ex  3 > 1 4
 135.44: ln (x + 3) = 2
 135.45: 45
 135.46: 30
 135.47: 180
 135.48: 2
 135.49: 7 6
 135.50: 4 3
Solutions for Chapter 135: Law of Cosines
Full solutions for Algebra 2, Student Edition (MERRILL ALGEBRA 2)  1st Edition
ISBN: 9780078738302
Solutions for Chapter 135: Law of Cosines
Get Full SolutionsChapter 135: Law of Cosines includes 50 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Algebra 2, Student Edition (MERRILL ALGEBRA 2) was written by and is associated to the ISBN: 9780078738302. Since 50 problems in chapter 135: Law of Cosines have been answered, more than 60204 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Algebra 2, Student Edition (MERRILL ALGEBRA 2), edition: 1.

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

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.

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

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

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.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

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 .

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

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.

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).

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.

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 ().