- 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
Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.
Upper triangular systems are solved in reverse order Xn to Xl.
peA) = det(A - AI) has peA) = zero matrix.
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 IIA-1 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 A-I.
Square matrix with A-I A = I and AA-l = 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 B-1 A-I and (A-I)T. Cofactor formula (A-l)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.
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).
If N NT = NT N, then N has orthonormal (complex) eigenvectors.
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 A-I are BT AT and (AT)-I.
Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.
Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).