 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
ISBN: 9781111826857
Solutions for Chapter 5: Identities and Formulas
Get Full SolutionsTrigonometry was written by and is associated to the ISBN: 9781111826857. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Trigonometry, edition: 7. Since 30 problems in chapter 5: Identities and Formulas have been answered, more than 25734 students have viewed full stepbystep solutions from this chapter. Chapter 5: Identities and Formulas includes 30 full stepbystep solutions.

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.

Cyclic shift
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.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + 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, offdiagonal entries = 0.

Incidence matrix of a directed graph.
The m by n edgenode 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)ยท(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.

Orthogonal subspaces.
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+ (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).

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 = Q1 = Q.

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!