 4.7.4.1.365: Graph y = cos x for x between 2n and 2n, and then reflect the grap...
 4.7.4.1.366: Graph y = sin x for x between n12 and n12, and then reflect the gr...
 4.7.4.1.367: Graph y tan x for x between  nl2 and n/2, and then reflect the gra...
 4.7.4.1.368: Graph y cot x for x between 0 and n, and then reflect the graph abo...
 4.7.4.1.369: Evaluate each expression without using a calculator, and write your...
 4.7.4.1.370: Evaluate each expression without using a calculator, and write your...
 4.7.4.1.371: Evaluate each expression without using a calculator, and write your...
 4.7.4.1.372: Evaluate each expression without using a calculator, and write your...
 4.7.4.1.373: Evaluate each expression without using a calculator, and write your...
 4.7.4.1.374: Evaluate each expression without using a calculator, and write your...
 4.7.4.1.375: Evaluate each expression without using a calculator, and write your...
 4.7.4.1.376: Evaluate each expression without using a calculator, and write your...
 4.7.4.1.377: Evaluate each expression without using a calculator, and write your...
 4.7.4.1.378: Evaluate each expression without using a calculator, and write your...
 4.7.4.1.379: Evaluate each expression without using a calculator, and write your...
 4.7.4.1.380: Evaluate each expression without using a calculator, and write your...
 4.7.4.1.381: Evaluate each expression without using a calculator, and write your...
 4.7.4.1.382: Evaluate each expression without using a calculator, and write your...
 4.7.4.1.383: Evaluate each expression without using a calculator, and write your...
 4.7.4.1.384: Evaluate each expression without using a calculator, and write your...
 4.7.4.1.385: Evaluate each expression without using a calculator, and write your...
 4.7.4.1.386: Evaluate each expression without using a calculator, and write your...
 4.7.4.1.387: Evaluate each expression without using a calculator, and write your...
 4.7.4.1.388: Evaluate each expression without using a calculator, and write your...
 4.7.4.1.389: Use a calculator to evaluate each expression to the nearest tenth o...
 4.7.4.1.390: Use a calculator to evaluate each expression to the nearest tenth o...
 4.7.4.1.391: Use a calculator to evaluate each expression to the nearest tenth o...
 4.7.4.1.392: Use a calculator to evaluate each expression to the nearest tenth o...
 4.7.4.1.393: Use a calculator to evaluate each expression to the nearest tenth o...
 4.7.4.1.394: Use a calculator to evaluate each expression to the nearest tenth o...
 4.7.4.1.395: Use a calculator to evaluate each expression to the nearest tenth o...
 4.7.4.1.396: Use a calculator to evaluate each expression to the nearest tenth o...
 4.7.4.1.397: Use a calculator to evaluate each expression to the nearest tenth o...
 4.7.4.1.398: Use a calculator to evaluate each expression to the nearest tenth o...
 4.7.4.1.399: Use a calculator to evaluate each expression to the nearest tenth o...
 4.7.4.1.400: Use a calculator to evaluate each expression to the nearest tenth o...
 4.7.4.1.401: Use a calculator to evaluate each expression to the nearest tenth o...
 4.7.4.1.402: Use a calculator to evaluate each expression to the nearest tenth o...
 4.7.4.1.403: Use your graphing calculator to graph y = sin 1 x in degree mode. ...
 4.7.4.1.404: Use your graphing calculator to graph y cosI x in degree mode. Use...
 4.7.4.1.405: Use your graphing calculator to graph y tanI x in degree mode. Use...
 4.7.4.1.406: Simplify 4 Icos (J if (J = sin1 =for some real number x.
 4.7.4.1.407: Simplify 2isin (JI if (J = cos ; for some real number x.
 4.7.4.1.408: Simplify 5 Isec (J Iif (J = tanI ~ for some real number x.
 4.7.4.1.409: Evaluate without using a calculator. sin (sin  1 t)
 4.7.4.1.410: Evaluate without using a calculator. cos (cos 1 ~)
 4.7.4.1.411: Evaluate without using a calculator. cos (COSI ~)
 4.7.4.1.412: Evaluate without using a calculator. sin (sin 1
 4.7.4.1.413: Evaluate without using a calculator. tan (tan 1 ~)
 4.7.4.1.414: Evaluate without using a calculator. tan (tanI! )
 4.7.4.1.415: Evaluate without using a calculator. sin 1 (sin 225)
 4.7.4.1.416: Evaluate without using a calculator. sin 1 (sin 330)
 4.7.4.1.417: Evaluate without using a calculator. sinI (sin ;
 4.7.4.1.418: Evaluate without using a calculator. sin I (sin :)
 4.7.4.1.419: Evaluate without using a calculator.
 4.7.4.1.420: Evaluate without using a calculator.
 4.7.4.1.421: Evaluate without using a calculator. COS cos 4
 4.7.4.1.422: Evaluate without using a calculator. cos 1 (cos 7;)
 4.7.4.1.423: Evaluate without using a calculator.
 4.7.4.1.424: Evaluate without using a calculator. tan 1 (tan 60)
 4.7.4.1.425: Evaluate without using a calculator. tanl (tan 5;)
 4.7.4.1.426: Evaluate without using a calculator. tanl (tan 2;)
 4.7.4.1.427: Evaluate without using a calculator. cos (tanI i)
 4.7.4.1.428: Evaluate without using a calculator. csc (tanl i)
 4.7.4.1.429: Evaluate without using a calculator. tan (sin  I ~
 4.7.4.1.430: Evaluate without using a calculator. tan (cos 1 f)
 4.7.4.1.431: Evaluate without using a calculator. sec (COS ~)
 4.7.4.1.432: Evaluate without using a calculator. sin (cos 1 ~
 4.7.4.1.433: Evaluate without using a calculator. sin (COS ~
 4.7.4.1.434: Evaluate without using a calculator. cos (Sin 1 t)
 4.7.4.1.435: Evaluate without using a calculator. cot (tanI ~)
 4.7.4.1.436: Evaluate without using a calculator. cot (tan + )
 4.7.4.1.437: Simplify sinI (sin x) if 7T/2 x::; 7T/2.
 4.7.4.1.438: Simplify cosI (cos x) if 0 ::; x::; 7T,
 4.7.4.1.439: For each expression below, write an equivalent expression that invo...
 4.7.4.1.440: For each expression below, write an equivalent expression that invo...
 4.7.4.1.441: For each expression below, write an equivalent expression that invo...
 4.7.4.1.442: For each expression below, write an equivalent expression that invo...
 4.7.4.1.443: For each expression below, write an equivalent expression that invo...
 4.7.4.1.444: For each expression below, write an equivalent expression that invo...
 4.7.4.1.445: For each expression below, write an equivalent expression that invo...
 4.7.4.1.446: For each expression below, write an equivalent expression that invo...
 4.7.4.1.447: For each expression below, write an equivalent expression that invo...
 4.7.4.1.448: For each expression below, write an equivalent expression that invo...
 4.7.4.1.449: Navigation The great circle distance between two points P1(LTt, LNI...
 4.7.4.1.450: Navigation The great circle distance between two points P1(LTt, LNI...
 4.7.4.1.451: The problems that follow review material we covered in Sections 4.2...
 4.7.4.1.452: The problems that follow review material we covered in Sections 4.2...
 4.7.4.1.453: The problems that follow review material we covered in Sections 4.2...
 4.7.4.1.454: The problems that follow review material we covered in Sections 4.2...
 4.7.4.1.455: The problems that follow review material we covered in Sections 4.2...
 4.7.4.1.456: The problems that follow review material we covered in Sections 4.2...
 4.7.4.1.457: Graph one complete cycle ofeach of the following equations. Be sure...
 4.7.4.1.458: Graph one complete cycle ofeach of the following equations. Be sure...
 4.7.4.1.459: Graph one complete cycle ofeach of the following equations. Be sure...
 4.7.4.1.460: Graph one complete cycle ofeach of the following equations. Be sure...
 4.7.4.1.461: Graph one complete cycle ofeach of the following equations. Be sure...
 4.7.4.1.462: Graph one complete cycle ofeach of the following equations. Be sure...
Solutions for Chapter 4.7: Graphing and Inverse Functions
Full solutions for Trigonometry
ISBN: 9780495108351
Solutions for Chapter 4.7: Graphing and Inverse Functions
Get Full SolutionsChapter 4.7: Graphing and Inverse Functions includes 98 full stepbystep solutions. This textbook survival guide was created for the textbook: Trigonometry, edition: . Trigonometry was written by and is associated to the ISBN: 9780495108351. This expansive textbook survival guide covers the following chapters and their solutions. Since 98 problems in chapter 4.7: Graphing and Inverse Functions have been answered, more than 29919 students have viewed full stepbystep solutions from this chapter.

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

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex 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.

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

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

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

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

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

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

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

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).

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

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.