 6.1.6.1.1: For each of the following equations, solve for (a) all degree solut...
 6.1.6.1.2: For each of the following equations, solve for (a) all degree solut...
 6.1.6.1.3: For each of the following equations, solve for (a) all degree solut...
 6.1.6.1.4: For each of the following equations, solve for (a) all degree solut...
 6.1.6.1.5: For each of the following equations, solve for (a) all degree solut...
 6.1.6.1.6: For each of the following equations, solve for (a) all degree solut...
 6.1.6.1.7: For each of the following equations, solve for (a) all radian solut...
 6.1.6.1.8: For each of the following equations, solve for (a) all radian solut...
 6.1.6.1.9: For each of the following equations, solve for (a) all radian solut...
 6.1.6.1.10: For each of the following equations, solve for (a) all radian solut...
 6.1.6.1.11: For each of the following equations, solve for (a) all radian solut...
 6.1.6.1.12: For each of the following equations, solve for (a) all radian solut...
 6.1.6.1.13: For each of the following equations, solve for (a) all degree solut...
 6.1.6.1.14: For each of the following equations, solve for (a) all degree solut...
 6.1.6.1.15: For each of the following equations, solve for (a) all degree solut...
 6.1.6.1.16: For each of the following equations, solve for (a) all degree solut...
 6.1.6.1.17: For each of the following equations, solve for (a) all degree solut...
 6.1.6.1.18: For each of the following equations, solve for (a) all degree solut...
 6.1.6.1.19: For each of the following equations, solve for (a) all radian solut...
 6.1.6.1.20: For each of the following equations, solve for (a) all radian solut...
 6.1.6.1.21: For each of the following equations, solve for (a) all radian solut...
 6.1.6.1.22: For each of the following equations, solve for (a) all radian solut...
 6.1.6.1.23: For each of the following equations, solve for (a) all radian solut...
 6.1.6.1.24: For each of the following equations, solve for (a) all radian solut...
 6.1.6.1.25: For each of the following equations, solve for (a) all radian solut...
 6.1.6.1.26: For each of the following equations, solve for (a) all radian solut...
 6.1.6.1.27: For each of the following equations, solve for (a) all degree solut...
 6.1.6.1.28: For each of the following equations, solve for (a) all degree solut...
 6.1.6.1.29: For each of the following equations, solve for (a) all degree solut...
 6.1.6.1.30: For each of the following equations, solve for (a) all degree solut...
 6.1.6.1.31: For each of the following equations, solve for (a) all degree solut...
 6.1.6.1.32: For each of the following equations, solve for (a) all degree solut...
 6.1.6.1.33: Use the quadratic formula to find (a) all degree solutions and (b) ...
 6.1.6.1.34: Use the quadratic formula to find (a) all degree solutions and (b) ...
 6.1.6.1.35: Use the quadratic formula to find (a) all degree solutions and (b) ...
 6.1.6.1.36: Use the quadratic formula to find (a) all degree solutions and (b) ...
 6.1.6.1.37: Use the quadratic formula to find (a) all degree solutions and (b) ...
 6.1.6.1.38: Use the quadratic formula to find (a) all degree solutions and (b) ...
 6.1.6.1.39: Find all degree solutions to the following equations.cos (A  50) = 2
 6.1.6.1.40: Find all degree solutions to the following equations.sin (A + 50) V3 2
 6.1.6.1.41: Find all degree solutions to the following equations.sin (A + 30) 2
 6.1.6.1.42: Find all degree solutions to the following equations.cos (A + 30) = 2
 6.1.6.1.43: Find all radian solutions to the following equations. COS(A ;)
 6.1.6.1.44: Find all radian solutions to the following equations. sin (A  ;) =...
 6.1.6.1.45: Find all radian solutions to the following equations. sin (A + ~)
 6.1.6.1.46: Find all radian solutions to the following equations. cos (A + )
 6.1.6.1.47: Use your graphing calculator to find the solutions to the equations...
 6.1.6.1.48: Use your graphing calculator to find the solutions to the equations...
 6.1.6.1.49: Use your graphing calculator to find the solutions to the equations...
 6.1.6.1.50: Use your graphing calculator to find the solutions to the equations...
 6.1.6.1.51: Use your graphing calculator to find the solutions to the equations...
 6.1.6.1.52: Use your graphing calculator to find the solutions to the equations...
 6.1.6.1.53: Use your graphing calculator to find the solutions to the equations...
 6.1.6.1.54: Use your graphing calculator to find the solutions to the equations...
 6.1.6.1.55: Use your graphing calculator to find the solutions to the equations...
 6.1.6.1.56: Use your graphing calculator to find the solutions to the equations...
 6.1.6.1.57: Use your graphing calculator to find the solutions to the equations...
 6.1.6.1.58: Use your graphing calculator to find the solutions to the equations...
 6.1.6.1.59: Use your graphing calculator to find the solutions to the equations...
 6.1.6.1.60: Use your graphing calculator to find the solutions to the equations...
 6.1.6.1.61: Use your graphing calculator to find the solutions to the equations...
 6.1.6.1.62: Use your graphing calculator to find the solutions to the equations...
 6.1.6.1.63: Use your graphing calculator to find the solutions to the equations...
 6.1.6.1.64: Use your graphing calculator to find the solutions to the equations...
 6.1.6.1.65: Give the equation for the height, if v is 1,500 feet per second and...
 6.1.6.1.66: Give the equation for h, if v is 600 feet per second and (J is 45. ...
 6.1.6.1.67: Use the equation found in to find the height of the object after 2 ...
 6.1.6.1.68: Use the equation found in to find the height of the object after V3...
 6.1.6.1.69: Find the angle of elevation 8 of a rifle barrel, if a bullet fired ...
 6.1.6.1.70: Find the angle of elevation of a rifle, if a bullet fired at 1,500 ...
 6.1.6.1.71: The problems that follow review material we covered in Sections 5.2...
 6.1.6.1.72: The problems that follow review material we covered in Sections 5.2...
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 6.1.6.1.79: The problems that follow review material we covered in Sections 5.2...
 6.1.6.1.80: The problems that follow review material we covered in Sections 5.2...
Solutions for Chapter 6.1: Equations
Full solutions for Trigonometry
ISBN: 9780495108351
Solutions for Chapter 6.1: Equations
Get Full SolutionsThis textbook survival guide was created for the textbook: Trigonometry, edition: . This expansive textbook survival guide covers the following chapters and their solutions. Since 80 problems in chapter 6.1: Equations have been answered, more than 32096 students have viewed full stepbystep solutions from this chapter. Chapter 6.1: Equations includes 80 full stepbystep solutions. Trigonometry was written by and is associated to the ISBN: 9780495108351.

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

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

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

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

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

Iterative method.
A sequence of steps intended to approach the desired solution.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

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

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

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.

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

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

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).

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.