 6.4.1: Exer. 16: Find the reference angle uR if u has the given measure.
 6.4.2: Exer. 16: Find the reference angle uR if u has the given measure.
 6.4.3: Exer. 16: Find the reference angle uR if u has the given measure.
 6.4.4: Exer. 16: Find the reference angle uR if u has the given measure.
 6.4.5: Exer. 16: Find the reference angle uR if u has the given measure.
 6.4.6: Exer. 16: Find the reference angle uR if u has the given measure.
 6.4.7: Exer. 718: Find the exact value.
 6.4.8: Exer. 718: Find the exact value.
 6.4.9: Exer. 718: Find the exact value.
 6.4.10: Exer. 718: Find the exact value.
 6.4.11: Exer. 718: Find the exact value.
 6.4.12: Exer. 718: Find the exact value.
 6.4.13: Exer. 718: Find the exact value.
 6.4.14: Exer. 718: Find the exact value.
 6.4.15: Exer. 718: Find the exact value.
 6.4.16: Exer. 718: Find the exact value.
 6.4.17: Exer. 718: Find the exact value.
 6.4.18: Exer. 718: Find the exact value.
 6.4.19: Exer. 1924: Approximate to three decimal places.
 6.4.20: Exer. 1924: Approximate to three decimal places.
 6.4.21: Exer. 1924: Approximate to three decimal places.
 6.4.22: Exer. 1924: Approximate to three decimal places.
 6.4.23: Exer. 1924: Approximate to three decimal places.
 6.4.24: Exer. 1924: Approximate to three decimal places.
 6.4.25: Exer. 2532: Approximate the acute angle u to the nearest (a) 0.01 a...
 6.4.26: Exer. 2532: Approximate the acute angle u to the nearest (a) 0.01 a...
 6.4.27: Exer. 2532: Approximate the acute angle u to the nearest (a) 0.01 a...
 6.4.28: Exer. 2532: Approximate the acute angle u to the nearest (a) 0.01 a...
 6.4.29: Exer. 2532: Approximate the acute angle u to the nearest (a) 0.01 a...
 6.4.30: Exer. 2532: Approximate the acute angle u to the nearest (a) 0.01 a...
 6.4.31: Exer. 2532: Approximate the acute angle u to the nearest (a) 0.01 a...
 6.4.32: Exer. 2532: Approximate the acute angle u to the nearest (a) 0.01 a...
 6.4.33: Exer. 3334: Approximate to four decimal places.
 6.4.34: Exer. 3334: Approximate to four decimal places.
 6.4.35: Exer. 3536: Approximate, to the nearest 0.1 , all angles u in the i...
 6.4.36: Exer. 3536: Approximate, to the nearest 0.1 , all angles u in the i...
 6.4.37: Exer. 3738: Approximate, to the nearest 0.01 radian, all angles u i...
 6.4.38: Exer. 3738: Approximate, to the nearest 0.01 radian, all angles u i...
 6.4.39: The thickness of the ozone layer can be estimated using the formula...
 6.4.40: Refer to Exercise 39. If the ozone layer is estimated to be 0.31 ce...
 6.4.41: The amount of sunshine illuminating a wall of a building can greatl...
 6.4.42: In the midlatitudes it is sometimes possible to estimate the dista...
 6.4.43: Points on the terminal sides of angles play an important part in th...
 6.4.44: Suppose the robots arm in Exercise 43 can change its length in addi...
Solutions for Chapter 6.4: Values of the Trigonometric Functions
Full solutions for Algebra and Trigonometry with Analytic Geometry  12th Edition
ISBN: 9780495559719
Solutions for Chapter 6.4: Values of the Trigonometric Functions
Get Full SolutionsSince 44 problems in chapter 6.4: Values of the Trigonometric Functions have been answered, more than 37702 students have viewed full stepbystep solutions from this chapter. Chapter 6.4: Values of the Trigonometric Functions includes 44 full stepbystep solutions. This textbook survival guide was created for the textbook: Algebra and Trigonometry with Analytic Geometry, edition: 12. This expansive textbook survival guide covers the following chapters and their solutions. Algebra and Trigonometry with Analytic Geometry was written by and is associated to the ISBN: 9780495559719.

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

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.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

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

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

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.

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 •

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

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.