- 13-5.1: B A C 35
- 13-5.2: A C 70 40
- 13-5.3: A 42, b 57, a 63 4
- 13-5.4: a 5, b 12, c 13
- 13-5.5: Find the distance from the pitchers mound to first base.
- 13-5.6: Find the angle between home plate, thepitchers mound, and first base.
- 13-5.7: 72 48 13
- 13-5.8: 15 19 18
- 13-5.9: 185 140 1
- 13-5.10: 34 15 12
- 13-5.11: 17 11
- 13-5.12: 71
- 13-5.13: a 20, c 24, B 47
- 13-5.14: a 345, b 648, c 442
- 13-5.15: A 36, a 10 , b 19
- 13-5.16: A 25, B 78, a 13.7
- 13-5.17: a 21.5, b 16.7, c 10.3
- 13-5.18: a 16, b 24, c 41
- 13-5.19: GEOMETRY In rhombus ABCD, the measure of ADC is 52. Find the measur...
- 13-5.20: SURVEYING Two sides of a triangular plot of land have lengths of 42...
- 13-5.21: a 8, b 24, c 18
- 13-5.22: B 19, a 51, c 61
- 13-5.23: A 56, B 22, a 12.2
- 13-5.24: a 4, b 8, c 5
- 13-5.25: a 21.5, b 13, C 38
- 13-5.26: A 40, b 7, a 6
- 13-5.27: An anthropologist examining the footprints made by a bipedal (two-f...
- 13-5.28: Find the step angle made by the hindfeet of a herbivorous dinosaur ...
- 13-5.29: An efficient walker has a step angle that approaches 180, meaning t...
- 13-5.30: AVIATION A pilot typically flies a route from Bloomington to Rockfo...
- 13-5.31: REASONING Explain how to solve a triangle by using the Law of Cosin...
- 13-5.32: FIND THE ERROR Mateo and Amy are deciding which method, the Law of ...
- 13-5.33: OPEN ENDED Give an example of a triangle that can be solved by firs...
- 13-5.34: CHALLENGE Explain how the Pythagorean Theorem is a special case of ...
- 13-5.35: Writing in Math Use the information on page 793 to explain how you ...
- 13-5.36: ACT/SAT In DEF, what is the value of to the nearest degree? A 26 B ...
- 13-5.37: REVIEW Two trucks, A and B, start from the intersection C of two st...
- 13-5.38: SANDBOX Mr. Blackwell is building a triangular sandbox. He is to jo...
- 13-5.39: (5, 12)
- 13-5.40: (4, 7)
- 13-5.41: (10 , 6 )
- 13-5.42: ex + 5 = 9 4
- 13-5.43: 4ex - 3 > -1 4
- 13-5.44: ln (x + 3) = 2
- 13-5.45: 45
- 13-5.46: 30
- 13-5.47: 180
- 13-5.48: 2
- 13-5.49: 7 6
- 13-5.50: 4 3
Solutions for Chapter 13-5: Law of Cosines
Full solutions for Algebra 2, Student Edition (MERRILL ALGEBRA 2) | 1st Edition
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.
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 A-I.
Square matrix with A-I A = I and AA-l = 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 B-1 A-I and (A-I)T. Cofactor formula (A-l)ij = Cji! detA.
lA-II = 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 .
= Xl (column 1) + ... + xn(column n) = combination of columns.
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+ (Moore-Penrose 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.
R = [~ CS ] rotates the plane by () and R- 1 = RT rotates back by -(). Eigenvalues are eiO and e-iO , eigenvectors are (1, ±i). c, s = cos (), sin ().