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# Solutions for Chapter 3.5: Radian Measure

## Full solutions for Trigonometry

ISBN: 9780495108351

Solutions for Chapter 3.5: Radian Measure

Solutions for Chapter 3.5
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##### ISBN: 9780495108351

Since 71 problems in chapter 3.5: Radian Measure have been answered, more than 32203 students have viewed full step-by-step solutions from this chapter. Trigonometry was written by and is associated to the ISBN: 9780495108351. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Trigonometry, edition: . Chapter 3.5: Radian Measure includes 71 full step-by-step solutions.

Key Math Terms and definitions covered in this textbook
• Affine transformation

Tv = Av + Vo = linear transformation plus shift.

• Back substitution.

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

• Circulant matrix C.

Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn - l . Cx = convolution c * x. Eigenvectors in F.

• Column picture of Ax = b.

The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

• Complex conjugate

z = a - ib for any complex number z = a + ib. Then zz = Iz12.

A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax - x Tb over growing Krylov subspaces.

• 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.

• Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.

Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

• Eigenvalue A and eigenvector x.

Ax = AX with x#-O so det(A - AI) = o.

• 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

• Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).

Use AT for complex A.

• Least squares solution X.

The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b - Ax is orthogonal to all columns of A.

• Length II x II.

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

• Lucas numbers

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.

• Markov matrix M.

All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

• Multiplication Ax

= Xl (column 1) + ... + xn(column n) = combination of columns.

• Rank r (A)

= number of pivots = dimension of column space = dimension of row space.

• Singular matrix A.

A square matrix that has no inverse: det(A) = o.

• Special solutions to As = O.

One free variable is Si = 1, other free variables = o.

• Tridiagonal matrix T: tij = 0 if Ii - j I > 1.

T- 1 has rank 1 above and below diagonal.

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