 5.6.1: Determine whether the positive or negative square root should be se...
 5.6.2: Determine whether the positive or negative square root should be se...
 5.6.3: Determine whether the positive or negative square root should be se...
 5.6.4: Determine whether the positive or negative square root should be se...
 5.6.5: Match each expression in Column I with its value in Column II. sin ...
 5.6.6: Match each expression in Column I with its value in Column II. tan ...
 5.6.7: Match each expression in Column I with its value in Column II. cos ...
 5.6.8: Match each expression in Column I with its value in Column II. tan ...
 5.6.9: Match each expression in Column I with its value in Column II. tan ...
 5.6.10: Match each expression in Column I with its value in Column II. cos ...
 5.6.11: Use a halfangle identity to find each exact value. See Examples 1 ...
 5.6.12: Use a halfangle identity to find each exact value. See Examples 1 ...
 5.6.13: Use a halfangle identity to find each exact value. See Examples 1 ...
 5.6.14: Use a halfangle identity to find each exact value. See Examples 1 ...
 5.6.15: Use a halfangle identity to find each exact value. See Examples 1 ...
 5.6.16: Use a halfangle identity to find each exact value. See Examples 1 ...
 5.6.17: Explain how to use identities from this section to find the exact v...
 5.6.18: The halfangle identity tan A 2 = {B 1  cos A 1 + cos A can be use...
 5.6.19: Use the given information to find each of the following. See Exampl...
 5.6.20: Use the given information to find each of the following. See Exampl...
 5.6.21: Use the given information to find each of the following. See Exampl...
 5.6.22: Use the given information to find each of the following. See Exampl...
 5.6.23: Use the given information to find each of the following. See Exampl...
 5.6.24: Use the given information to find each of the following. See Exampl...
 5.6.25: Use the given information to find each of the following. See Exampl...
 5.6.26: Use the given information to find each of the following. See Exampl...
 5.6.27: Use the given information to find each of the following. See Exampl...
 5.6.28: Use the given information to find each of the following. See Exampl...
 5.6.29: Use the given information to find each of the following. See Exampl...
 5.6.30: Use the given information to find each of the following. See Exampl...
 5.6.31: Concept Check If cos x 0.9682 and sin x = 0.250, then tan x 2 .
 5.6.32: Concept Check If cos x = 0.750 and sin x 0.6614, then tan x 2 .
 5.6.33: Simplify each expression. See Example 4. B 1  cos 40 2
 5.6.34: Simplify each expression. See Example 4. B 1 + cos 76 2
 5.6.35: Simplify each expression. See Example 4. B 1  cos 147 1 + cos 147
 5.6.36: Simplify each expression. See Example 4. B 1 + cos 165 1  cos 165
 5.6.37: Simplify each expression. See Example 4. 1  cos 59.74 sin 59.74
 5.6.38: Simplify each expression. See Example 4. sin 158.2 1 + cos 158.2
 5.6.39: Simplify each expression. See Example 4. {B 1 + cos 18x 2
 5.6.40: Simplify each expression. See Example 4. {B 1 + cos 20a 2
 5.6.41: Simplify each expression. See Example 4. {B 1  cos 8u 1 + cos 8u
 5.6.42: Simplify each expression. See Example 4. {B 1  cos 5A 1 + cos 5A
 5.6.43: Simplify each expression. See Example 4. {B 1 + cos x 4 2
 5.6.44: Simplify each expression. See Example 4. { D 1  cos 3u 5 2
 5.6.45: Verify that each equation is an identity. See Example 5. sec2 x 2 =...
 5.6.46: Verify that each equation is an identity. See Example 5. cot2 x 2 =...
 5.6.47: Verify that each equation is an identity. See Example 5. sin2 x 2 =...
 5.6.48: Verify that each equation is an identity. See Example 5. sin 2x 2 s...
 5.6.49: Verify that each equation is an identity. See Example 5. 2 1 + cos ...
 5.6.50: Verify that each equation is an identity. See Example 5. tan u 2 = ...
 5.6.51: Verify that each equation is an identity. See Example 5. 1  tan2 u...
 5.6.52: Verify that each equation is an identity. See Example 5. cos x = 1 ...
 5.6.53: Use the halfangle identity tan A 2 = sin A 1 + cos A to derive the...
 5.6.54: Use the identity tan A 2 = sin A 1 + cos A to determine an identity...
 5.6.55: Graph each expression and use the graph to make a conjecture, predi...
 5.6.56: Graph each expression and use the graph to make a conjecture, predi...
 5.6.57: Graph each expression and use the graph to make a conjecture, predi...
 5.6.58: Graph each expression and use the graph to make a conjecture, predi...
 5.6.59: An airplane flying faster than the speed of sound sends out sound w...
 5.6.60: An airplane flying faster than the speed of sound sends out sound w...
 5.6.61: An airplane flying faster than the speed of sound sends out sound w...
 5.6.62: An airplane flying faster than the speed of sound sends out sound w...
 5.6.63: (Modeling) Railroad Curves In the United States, circular railroad ...
 5.6.64: In Exercise 63, if b = 12, what is the measure of angle u to the ne...
 5.6.65: (See Hobsons A Treatise on Plane Trigonometry.) Use this value and ...
 5.6.66: (See Hobsons A Treatise on Plane Trigonometry.) Use this value and ...
 5.6.67: (See Hobsons A Treatise on Plane Trigonometry.) Use this value and ...
 5.6.68: (See Hobsons A Treatise on Plane Trigonometry.) Use this value and ...
 5.6.69: (See Hobsons A Treatise on Plane Trigonometry.) Use this value and ...
 5.6.70: (See Hobsons A Treatise on Plane Trigonometry.) Use this value and ...
 5.6.71: (See Hobsons A Treatise on Plane Trigonometry.) Use this value and ...
 5.6.72: (See Hobsons A Treatise on Plane Trigonometry.) Use this value and ...
 5.6.73: (See Hobsons A Treatise on Plane Trigonometry.) Use this value and ...
 5.6.74: (See Hobsons A Treatise on Plane Trigonometry.) Use this value and ...
 5.6.75: (See Hobsons A Treatise on Plane Trigonometry.) Use this value and ...
 5.6.76: (See Hobsons A Treatise on Plane Trigonometry.) Use this value and ...
 5.6.77: These exercises use results from plane geometry to obtain exact val...
 5.6.78: These exercises use results from plane geometry to obtain exact val...
 5.6.79: These exercises use results from plane geometry to obtain exact val...
 5.6.80: These exercises use results from plane geometry to obtain exact val...
 5.6.81: These exercises use results from plane geometry to obtain exact val...
 5.6.82: These exercises use results from plane geometry to obtain exact val...
 5.6.83: These exercises use results from plane geometry to obtain exact val...
 5.6.84: These exercises use results from plane geometry to obtain exact val...
Solutions for Chapter 5.6: HalfAngle Identities
Full solutions for Trigonometry  11th Edition
ISBN: 9780134217437
Solutions for Chapter 5.6: HalfAngle Identities
Get Full SolutionsThis textbook survival guide was created for the textbook: Trigonometry, edition: 11. Trigonometry was written by and is associated to the ISBN: 9780134217437. Since 84 problems in chapter 5.6: HalfAngle Identities have been answered, more than 24151 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 5.6: HalfAngle Identities includes 84 full stepbystep solutions.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

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.

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).

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

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

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.

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

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

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

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.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

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.

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.