 4.1: y sin x
 4.2: y cos x
 4.3: y tan x
 4.4: y sec x
 4.5: Show that cotangent is an odd function
 4.6: Prove the identity sin () sec () cot () 1. For e
 4.7: y cos x 8
 4.8: y 3 cos x
 4.9: y 2 3 sin 2x, x 210.
 4.10: y 2 sin x, 4 x 4 F
 4.11: y sin x 1
 4.12: y 3 sin 2x
 4.13: y 3 3 sin x 14.
 4.14: y 1 csc x
 4.15: y 3 tan 2x Gr
 4.16: y 2 sin (3x ), x 17
 4.17: y 2 sin x , x Fi
 4.18: y 2 sin x , x Fi
 4.19: y 2 sin x , x Fi
 4.20: y x sin x 2
 4.21: y sin x cos 2x
 4.22: y cos1 x 2
 4.23: y arcsin x
 4.24: sin1 2
 4.25: arctan (1)
 4.26: arcsin (0.5934)
 4.27: arctan (0.8302)
 4.28: tan cos1 2
 4.29: tan1 tan
 4.30: Write an equivalent algebraic expression for sin (cos1 x) that invo...
Solutions for Chapter 4: Graphing and Inverse Functions
Full solutions for Trigonometry  7th Edition
ISBN: 9781111826857
Solutions for Chapter 4: Graphing and Inverse Functions
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Chapter 4: Graphing and Inverse Functions includes 30 full stepbystep solutions. This textbook survival guide was created for the textbook: Trigonometry, edition: 7. Since 30 problems in chapter 4: Graphing and Inverse Functions have been answered, more than 24936 students have viewed full stepbystep solutions from this chapter. Trigonometry was written by and is associated to the ISBN: 9781111826857.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

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.

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

Inverse matrix AI.
Square matrix with AI A = I and AAl = 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 B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

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.

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

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

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).