- 3.1 - 3.3.1: Convert each degree measure to radians. 225
- 3.1 - 3.3.2: Convert each degree measure to radians. -330_
- 3.1 - 3.3.3: Convert each radian measure to degrees 5p 3
- 3.1 - 3.3.4: Convert each radian measure to degrees - 7p 6 A central angle of a ...
- 3.1 - 3.3.5: A central angle of a circle with radius 300 in. intercepts an arc o...
- 3.1 - 3.3.6: A central angle of a circle with radius 300 in. intercepts an arc o...
- 3.1 - 3.3.7: Find each circular function value. Give exact values. cos 7p 4
- 3.1 - 3.3.8: Find each circular function value. Give exact values. sin a- 5p 6 b
- 3.1 - 3.3.9: Find each circular function value. Give exact values. tan 3p
- 3.1 - 3.3.10: Find the exact value of s in the interval 3p2 , p4 if sin s = 23 2
Solutions for Chapter 3.1 - 3.3: Quiz
Full solutions for Trigonometry | 10th Edition
Upper triangular systems are solved in reverse order Xn to Xl.
Characteristic equation det(A - AI) = O.
The n roots are the eigenvalues of A.
Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.
Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.
Eigenvalue A and eigenvector x.
Ax = AX with x#-O so det(A - AI) = o.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.
Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.
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 Fn-1c can be computed with ne/2 multiplications. Revolutionary.
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 nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.
Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A - AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).
Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.
Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.
Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.
Row space C (AT) = all combinations of rows of A.
Column vectors by convention.
Symmetric matrix A.
The transpose is AT = A, and aU = a ji. A-I is also symmetric.
Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn- 1 with P(Xi) = bi. Vij = (Xi)j-I and det V = product of (Xk - Xi) for k > i.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.
Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.