 8.6.1: Concept Check Match the ordered pair from Column II with the pair o...
 8.6.2: Concept Check Match the ordered pair from Column II with the pair o...
 8.6.3: Concept Check Match the ordered pair from Column II with the pair o...
 8.6.4: Concept Check Match the ordered pair from Column II with the pair o...
 8.6.5: For each plane curve, (a) graph the curve, and (b) find a rectangul...
 8.6.6: For each plane curve, (a) graph the curve, and (b) find a rectangul...
 8.6.7: For each plane curve, (a) graph the curve, and (b) find a rectangul...
 8.6.8: For each plane curve, (a) graph the curve, and (b) find a rectangul...
 8.6.9: For each plane curve, (a) graph the curve, and (b) find a rectangul...
 8.6.10: For each plane curve, (a) graph the curve, and (b) find a rectangul...
 8.6.11: For each plane curve, (a) graph the curve, and (b) find a rectangul...
 8.6.12: For each plane curve, (a) graph the curve, and (b) find a rectangul...
 8.6.13: For each plane curve, (a) graph the curve, and (b) find a rectangul...
 8.6.14: For each plane curve, (a) graph the curve, and (b) find a rectangul...
 8.6.15: For each plane curve, (a) graph the curve, and (b) find a rectangul...
 8.6.16: For each plane curve, (a) graph the curve, and (b) find a rectangul...
 8.6.17: For each plane curve, (a) graph the curve, and (b) find a rectangul...
 8.6.18: For each plane curve, (a) graph the curve, and (b) find a rectangul...
 8.6.19: For each plane curve, (a) graph the curve, and (b) find a rectangul...
 8.6.20: For each plane curve, (a) graph the curve, and (b) find a rectangul...
 8.6.21: For each plane curve, (a) graph the curve, and (b) find a rectangul...
 8.6.22: For each plane curve, (a) graph the curve, and (b) find a rectangul...
 8.6.23: For each plane curve, (a) graph the curve, and (b) find a rectangul...
 8.6.24: For each plane curve, (a) graph the curve, and (b) find a rectangul...
 8.6.25: Graph each plane curve defined by the parametric equations for t in...
 8.6.26: Graph each plane curve defined by the parametric equations for t in...
 8.6.27: Graph each plane curve defined by the parametric equations for t in...
 8.6.28: Graph each plane curve defined by the parametric equations for t in...
 8.6.29: Give two parametric representations for the equation of each parabo...
 8.6.30: Give two parametric representations for the equation of each parabo...
 8.6.31: Give two parametric representations for the equation of each parabo...
 8.6.32: Give two parametric representations for the equation of each parabo...
 8.6.33: Graph each cycloid defined by the given equations for t in the spec...
 8.6.34: Graph each cycloid defined by the given equations for t in the spec...
 8.6.35: Lissajous Figures The screen shown here is an example of a Lissajou...
 8.6.36: Lissajous Figures The screen shown here is an example of a Lissajou...
 8.6.37: Lissajous Figures The screen shown here is an example of a Lissajou...
 8.6.38: Lissajous Figures The screen shown here is an example of a Lissajou...
 8.6.39: Flight of a Model Rocket A model rocket is launched from the ground...
 8.6.40: Flight of a Golf Ball Tyler is playing golf. He hits a golf ball fr...
 8.6.41: Flight of a Softball Sally hits a softball when it is 2 ft above th...
 8.6.42: Flight of a Baseball Carlos hits a baseball when it is 2.5 ft above...
 8.6.43: Path of a Rocket A rocket is launched from the top of an 8ft ladde...
 8.6.44: Simulating Gravity on the Moon If an object is thrown on the moon, ...
 8.6.45: Flight of a Baseball A baseball is hit from a height of 3 ft at a 6...
 8.6.46: (Modeling) Path of a Projectile In Exercises 46 and 47, a projectil...
 8.6.47: (Modeling) Path of a Projectile In Exercises 46 and 47, a projectil...
 8.6.48: Give two parametric representations of the line through the point 1...
 8.6.49: Give two parametric representations of the parabola y = a1x  h22 + k.
 8.6.50: Give a parametric representation of the rectangular equation x2 a2 ...
 8.6.51: Give a parametric representation of the rectangular equation x2 a2 ...
 8.6.52: The spiral of Archimedes has polar equation r = a u, where r2 = x2 ...
 8.6.53: Show that the hyperbolic spiral r u = a, where r2 = x2 + y2, is giv...
 8.6.54: The parametric equations x = cos t, y = sin t, for t in 30, 2p4 and...
 8.6.55: How is the graph affected if the equation x = 1t2 is replaced by x ...
 8.6.56: How is the graph affected if the equation y = g1t2 is replaced by y...
Solutions for Chapter 8.6: Parametric Equations, Graphs, and Applications
Full solutions for Trigonometry  10th Edition
ISBN: 9780321671776
Solutions for Chapter 8.6: Parametric Equations, Graphs, and Applications
Get Full SolutionsChapter 8.6: Parametric Equations, Graphs, and Applications includes 56 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 56 problems in chapter 8.6: Parametric Equations, Graphs, and Applications have been answered, more than 32728 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Trigonometry, edition: 10. Trigonometry was written by and is associated to the ISBN: 9780321671776.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

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

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

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.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Outer product uv T
= column times row = rank one matrix.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

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.

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

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.

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