 4.6.4.6.1: The graphs of the tangent, cotangent, secant, and cosecant function...
 4.6.4.6.2: To sketch the graph of a secant or cosecant function, first make a ...
 4.6.4.6.3: To sketch the graph of a secant or cosecant function, first make a ...
 4.6.4.6.4: In Exercises 14, use the graph of the function to answer the follow...
 4.6.4.6.5: In Exercises 524, sketch the graph of the function. (Include two fu...
 4.6.4.6.6: In Exercises 524, sketch the graph of the function. (Include two fu...
 4.6.4.6.7: In Exercises 524, sketch the graph of the function. (Include two fu...
 4.6.4.6.8: In Exercises 524, sketch the graph of the function. (Include two fu...
 4.6.4.6.9: In Exercises 524, sketch the graph of the function. (Include two fu...
 4.6.4.6.10: In Exercises 524, sketch the graph of the function. (Include two fu...
 4.6.4.6.11: In Exercises 524, sketch the graph of the function. (Include two fu...
 4.6.4.6.12: In Exercises 524, sketch the graph of the function. (Include two fu...
 4.6.4.6.13: In Exercises 524, sketch the graph of the function. (Include two fu...
 4.6.4.6.14: In Exercises 524, sketch the graph of the function. (Include two fu...
 4.6.4.6.15: In Exercises 524, sketch the graph of the function. (Include two fu...
 4.6.4.6.16: In Exercises 524, sketch the graph of the function. (Include two fu...
 4.6.4.6.17: In Exercises 524, sketch the graph of the function. (Include two fu...
 4.6.4.6.18: In Exercises 524, sketch the graph of the function. (Include two fu...
 4.6.4.6.19: In Exercises 524, sketch the graph of the function. (Include two fu...
 4.6.4.6.20: In Exercises 524, sketch the graph of the function. (Include two fu...
 4.6.4.6.21: In Exercises 524, sketch the graph of the function. (Include two fu...
 4.6.4.6.22: In Exercises 524, sketch the graph of the function. (Include two fu...
 4.6.4.6.23: In Exercises 524, sketch the graph of the function. (Include two fu...
 4.6.4.6.24: In Exercises 524, sketch the graph of the function. (Include two fu...
 4.6.4.6.25: In Exercises 2530, use a graphing utility to graph the function (in...
 4.6.4.6.26: In Exercises 2530, use a graphing utility to graph the function (in...
 4.6.4.6.27: In Exercises 2530, use a graphing utility to graph the function (in...
 4.6.4.6.28: In Exercises 2530, use a graphing utility to graph the function (in...
 4.6.4.6.29: In Exercises 2530, use a graphing utility to graph the function (in...
 4.6.4.6.30: In Exercises 2530, use a graphing utility to graph the function (in...
 4.6.4.6.31: In Exercises 3134, use a graph of the function to approximate the s...
 4.6.4.6.32: In Exercises 3134, use a graph of the function to approximate the s...
 4.6.4.6.33: In Exercises 3134, use a graph of the function to approximate the s...
 4.6.4.6.34: In Exercises 3134, use a graph of the function to approximate the s...
 4.6.4.6.35: In Exercises 3538, use the graph of the function to determine wheth...
 4.6.4.6.36: In Exercises 3538, use the graph of the function to determine wheth...
 4.6.4.6.37: In Exercises 3538, use the graph of the function to determine wheth...
 4.6.4.6.38: In Exercises 3538, use the graph of the function to determine wheth...
 4.6.4.6.39: In Exercises 3942, use a graphing utility to graph the two equation...
 4.6.4.6.40: In Exercises 3942, use a graphing utility to graph the two equation...
 4.6.4.6.41: In Exercises 3942, use a graphing utility to graph the two equation...
 4.6.4.6.42: In Exercises 3942, use a graphing utility to graph the two equation...
 4.6.4.6.43: In Exercises 4346, match the function with its graph. Describe the ...
 4.6.4.6.44: In Exercises 4346, match the function with its graph. Describe the ...
 4.6.4.6.45: In Exercises 4346, match the function with its graph. Describe the ...
 4.6.4.6.46: In Exercises 4346, match the function with its graph. Describe the ...
 4.6.4.6.47: Conjecture In Exercises 4750, use a graphing utility to graph the f...
 4.6.4.6.48: Conjecture In Exercises 4750, use a graphing utility to graph the f...
 4.6.4.6.49: Conjecture In Exercises 4750, use a graphing utility to graph the f...
 4.6.4.6.50: Conjecture In Exercises 4750, use a graphing utility to graph the f...
 4.6.4.6.51: In Exercises 5154, use a graphing utility to graph the function and...
 4.6.4.6.52: In Exercises 5154, use a graphing utility to graph the function and...
 4.6.4.6.53: In Exercises 5154, use a graphing utility to graph the function and...
 4.6.4.6.54: In Exercises 5154, use a graphing utility to graph the function and...
 4.6.4.6.55: Exploration In Exercises 55 and 56, use a graphing utility to graph...
 4.6.4.6.56: Exploration In Exercises 55 and 56, use a graphing utility to graph...
 4.6.4.6.57: Exploration In Exercises 57 and 58, use a graphing utility to graph...
 4.6.4.6.58: Exploration In Exercises 57 and 58, use a graphing utility to graph...
 4.6.4.6.59: PredatorPrey Model The population P of coyotes (a predator) at tim...
 4.6.4.6.60: Meteorology The normal monthly high temperatures (in degrees Fahren...
 4.6.4.6.61: Distance A plane flying at an altitude of 5 miles over level ground...
 4.6.4.6.62: Television Coverage A television camera is on a reviewing platform ...
 4.6.4.6.63: Harmonic Motion An object weighing pounds is suspended from a ceili...
 4.6.4.6.64: Numerical and Graphical Reasoning A crossed belt connects a 10cent...
 4.6.4.6.65: True or False? In Exercises 65 and 66, determine whether the statem...
 4.6.4.6.66: True or False? In Exercises 65 and 66, determine whether the statem...
 4.6.4.6.67: Graphical Reasoning Consider the functions and on the interval (a) ...
 4.6.4.6.68: Pattern Recognition (a) Use a graphing utility to graph each functi...
 4.6.4.6.69: In Exercises 69 and 70, use a graphing utility to explore the ratio...
 4.6.4.6.70: In Exercises 69 and 70, use a graphing utility to explore the ratio...
 4.6.4.6.71: In Exercises 71 and 72, determine which function is represented by ...
 4.6.4.6.72: In Exercises 71 and 72, determine which function is represented by ...
 4.6.4.6.73: Approximation Using calculus, it can be shown that the tangent func...
 4.6.4.6.74: Approximation Using calculus, it can be shown that the secant funct...
 4.6.4.6.75: In Exercises 7578, identify the rule of algebra illustrated by the ...
 4.6.4.6.76: In Exercises 7578, identify the rule of algebra illustrated by the ...
 4.6.4.6.77: In Exercises 7578, identify the rule of algebra illustrated by the ...
 4.6.4.6.78: In Exercises 7578, identify the rule of algebra illustrated by the ...
 4.6.4.6.79: In Exercises 7982, determine whether the function is onetoone. If...
 4.6.4.6.80: In Exercises 7982, determine whether the function is onetoone. If...
 4.6.4.6.81: In Exercises 7982, determine whether the function is onetoone. If...
 4.6.4.6.82: In Exercises 7982, determine whether the function is onetoone. If...
Solutions for Chapter 4.6: Graphs of Other Trigonometric Functions
Full solutions for Precalculus With Limits A Graphing Approach  5th Edition
ISBN: 9780618851522
Solutions for Chapter 4.6: Graphs of Other Trigonometric Functions
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Precalculus With Limits A Graphing Approach, edition: 5. Chapter 4.6: Graphs of Other Trigonometric Functions includes 82 full stepbystep solutions. Precalculus With Limits A Graphing Approach was written by and is associated to the ISBN: 9780618851522. Since 82 problems in chapter 4.6: Graphs of Other Trigonometric Functions have been answered, more than 33077 students have viewed full stepbystep solutions from this chapter.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

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.

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

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.

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.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)ยท(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 .

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

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.

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).

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 r (A)
= number of pivots = dimension of column space = dimension of row space.

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

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

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.