 5.3.5.1.171: Let sin A = with A in QIII and find sin 2A
 5.3.5.1.172: Let sin A = with A in QIII and find cos 2A
 5.3.5.1.173: Let sin A = with A in QIII and find tan 2A
 5.3.5.1.174: Let sin A = with A in QIII and find cot 2A
 5.3.5.1.175: Let cos x = lIVlO with x in QIV and find cos 2x
 5.3.5.1.176: Let cos x = lIVlO with x in QIV and find sin 2x
 5.3.5.1.177: Let cos x = lIVlO with x in QIV and find cot2x
 5.3.5.1.178: Let cos x = lIVlO with x in QIV and find tan 2x
 5.3.5.1.179: Let tan 0 = ;2 with 0 in QI and find sin 20
 5.3.5.1.180: Let tan 0 = ;2 with 0 in QI and find cos 28
 5.3.5.1.181: Let tan 0 = ;2 with 0 in QI and find csc 20
 5.3.5.1.182: Let tan 0 = ;2 with 0 in QI and find sec 28
 5.3.5.1.183: Let csc t = Vs with t in QII and find cos 2t
 5.3.5.1.184: Let csc t = Vs with t in QII and find sin 2t
 5.3.5.1.185: Let csc t = Vs with t in QII and find sec 2t
 5.3.5.1.186: Let csc t = Vs with t in QII and find csc 2t
 5.3.5.1.187: Graph each of the following from x = Oto x = 27T. Y 4  8 sin2 x
 5.3.5.1.188: Graph each of the following from x = Oto x = 27T. Y = 2 4 sin2 x
 5.3.5.1.189: Graph each of the following from x = Oto x = 27T. y = 6 cos2 X  3
 5.3.5.1.190: Graph each of the following from x = Oto x = 27T. y = 4 cos2 X  2
 5.3.5.1.191: Graph each of the following from x = Oto x = 27T. Y = 1 2 sin2 2x
 5.3.5.1.192: Graph each of the following from x = Oto x = 27T. Y = 2 cos2 2x 1
 5.3.5.1.193: Use exact values to show that each of the following is true. sin 60...
 5.3.5.1.194: Use exact values to show that each of the following is true. cos 60...
 5.3.5.1.195: Use exact values to show that each of the following is true. cos 12...
 5.3.5.1.196: Use exact values to show that each of the following is true. sin 90...
 5.3.5.1.197: Use exact values to show that each of the following is true. If tan...
 5.3.5.1.198: Use exact values to show that each of the following is true. If tan...
 5.3.5.1.199: Simplify each of the following. 2 sin 15 cos 15
 5.3.5.1.200: Simplify each of the following. cos2 15  sin2 15
 5.3.5.1.201: Simplify each of the following. 1 2 sin2 75
 5.3.5.1.202: Simplify each of the following. 2 cos2 105  1
 5.3.5.1.203: Simplify each of the following. sin cos 12 12
 5.3.5.1.204: Simplify each of the following. sm 8 cos 8
 5.3.5.1.205: Simplify each of the following. 1  tan2 22.5
 5.3.5.1.206: Simplify each of the following. tan8 37T
 5.3.5.1.207: Prove each of the following identities.(sin x cos xf = 1 sin 2x
 5.3.5.1.208: Prove each of the following identities.(cos x sin x)(cos x + sin x)...
 5.3.5.1.209: Prove each of the following identities.cos 0 = 2
 5.3.5.1.210: Prove each of the following identities.sin2 {} 2
 5.3.5.1.211: Prove each of the following identities.cot 8 1 cos 2{}
 5.3.5.1.212: Prove each of the following identities.cos 2{} = ;: 1 + 8
 5.3.5.1.213: Prove each of the following identities.2 csc 2x = tan x + cot x
 5.3.5.1.214: Prove each of the following identities.2 cot 2x = cot x  tan x
 5.3.5.1.215: Prove each of the following identities.sin 38 = 3 sin 8  4 sin3 8
 5.3.5.1.216: Prove each of the following identities.cos 38 = 4 cos3 8  3 cos 8
 5.3.5.1.217: Prove each of the following identities.cos4 x  sin4 x = cos 2x
 5.3.5.1.218: Prove each of the following identities.2 sin4 x + 2 sin2 x cos2 x =...
 5.3.5.1.219: Prove each of the following identities.cot 8  tan 8 =  sin 8 cos 8
 5.3.5.1.220: Prove each of the following identities.csc 8  2 sin 8 =  sin 8
 5.3.5.1.221: Prove each of the following identities.sin 4A = 4 sin A cos3 A  4 ...
 5.3.5.1.222: Prove each of the following identities.cos 4A = cos4 A  6 cos2 A s...
 5.3.5.1.223: Prove each of the following identities.1  tan x 1  sin 2x 1 + tan...
 5.3.5.1.224: Prove each of the following identities.2  2 cos 2x = sec x csc x ...
 5.3.5.1.225: Prove each of the following identities.Use your graphing calculator...
 5.3.5.1.226: Use your graphing calculator to determine if each equation appears ...
 5.3.5.1.227: Use your graphing calculator to determine if each equation appears ...
 5.3.5.1.228: Use your graphing calculator to determine if each equation appears ...
 5.3.5.1.229: Use your graphing calculator to determine if each equation appears ...
 5.3.5.1.230: Use your graphing calculator to determine if each equation appears ...
 5.3.5.1.231: Use your graphing calculator to determine if each equation appears ...
 5.3.5.1.232: Use your graphing calculator to determine if each equation appears ...
 5.3.5.1.233: Use your graphing calculator to determine if each equation appears ...
 5.3.5.1.234: Use your graphing calculator to determine if each equation appears ...
 5.3.5.1.235: The problems that follow review material we covered in Sections 4.3...
 5.3.5.1.236: The problems that follow review material we covered in Sections 4.3...
 5.3.5.1.237: The problems that follow review material we covered in Sections 4.3...
 5.3.5.1.238: The problems that follow review material we covered in Sections 4.3...
 5.3.5.1.239: The problems that follow review material we covered in Sections 4.3...
 5.3.5.1.240: The problems that follow review material we covered in Sections 4.3...
 5.3.5.1.241: Graph each of the following from x = 0 to x = 8. y = 2x + SIll 7TX
 5.3.5.1.242: Graph each of the following from x = 0 to x = 8. y = x+ SIll 2x
Solutions for Chapter 5.3: Identities and Formulas
Full solutions for Trigonometry
ISBN: 9780495108351
Solutions for Chapter 5.3: Identities and Formulas
Get Full SolutionsTrigonometry was written by and is associated to the ISBN: 9780495108351. This textbook survival guide was created for the textbook: Trigonometry, edition: . This expansive textbook survival guide covers the following chapters and their solutions. Chapter 5.3: Identities and Formulas includes 72 full stepbystep solutions. Since 72 problems in chapter 5.3: Identities and Formulas have been answered, more than 46381 students have viewed full stepbystep solutions from this chapter.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

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 Fn1c can be computed with ne/2 multiplications. Revolutionary.

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 .

Iterative method.
A sequence of steps intended to approach the desired solution.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

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 .

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.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

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

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.