- 5.1: c s o c t cos 2. (
- 5.2: (sec x 1)(sec x 1) tan2 x 3
- 5.3: sec cos tan sin 4. 1
- 5.4: cos si t n t 1 cos si t
- 5.5: 1 1 sin t 1 1 sin t 2 sec2 t 6
- 5.6: cos 2 sin 7.
- 5.7: cos4 A sin4 A cos 2A 8.
- 5.8: cot A 1 sin co 2 s A 2A
- 5.9: cot x tan x sin co x s c 2 o x s x 1
- 5.10: tan 2 x sec ta x n x 1
- 5.11: s s e in c 2 2 csc2 sec2
- 5.12: tan 1 cot csc sec Let
- 5.13: sin (A B)
- 5.14: cos (A B)
- 5.15: cos 2B
- 5.16: sin A 2
- 5.17: sin 75
- 5.18: tan 1 2
- 5.19: cos 4x cos 5x sin 4x sin 5x
- 5.20: sin 15 cos 75 cos 15 sin 75
- 5.21: If sin A with 180 A 270, find cos 2A and cos . 2
- 5.22: If sec A 10with 0 A 90, find sin 2A and sin . 2
- 5.23: Find tan A if tan B and tan (A B ) 3.
- 5.24: Find cos x if cos 2x .
- 5.25: cos arcsin 4 5 arctan 22
- 5.26: sin arccos 4 5 arctan 2
- 5.27: cos (2 sin1 x)
- 5.28: sin (2 cos1 x)
- 5.29: Rewrite the product sin 6x sin 4x as a sum or difference.
- 5.30: Rewrite the sum cos 15 cos 75 as a product and simplify
Solutions for Chapter 5: Identities and Formulas
Full solutions for Trigonometry | 7th Edition
Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).
Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!
Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.
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.
0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].
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.
Identity matrix I (or In).
Diagonal entries = 1, off-diagonal entries = 0.
Incidence matrix of a directed graph.
The m by n edge-node incidence matrix has a row for each edge (node i to node j), with entries -1 and 1 in columns i and j .
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.
Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = 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 .
Nullspace matrix N.
The columns of N are the n - r special solutions to As = O.
Every v in V is orthogonal to every w in W.
Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.
Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b - Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) -1 AT.
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).
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).
Reflection matrix (Householder) Q = I -2uuT.
Unit vector u is reflected to Qu = -u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q-1 = Q.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!