 7.6.1: cos12u2 = cos2 u  =  1 = 1  .
 7.6.2: sin2 u 2 = 2 .
 7.6.3: tan u 2 = 1  cos u .
 7.6.4: True or False tan12u2 = 2 tan u 1  tan2 u
 7.6.5: True or False sin12u2 has two equivalent forms: 2 sin u cos u and s...
 7.6.6: True or False tan12u2 + tan12u2 = tan14u2
 7.6.7: In 718, use the information given about the angle u, 0 u 6 2p, to f...
 7.6.8: In 718, use the information given about the angle u, 0 u 6 2p, to f...
 7.6.9: In 718, use the information given about the angle u, 0 u 6 2p, to f...
 7.6.10: In 718, use the information given about the angle u, 0 u 6 2p, to f...
 7.6.11: In 718, use the information given about the angle u, 0 u 6 2p, to f...
 7.6.12: In 718, use the information given about the angle u, 0 u 6 2p, to f...
 7.6.13: In 718, use the information given about the angle u, 0 u 6 2p, to f...
 7.6.14: In 718, use the information given about the angle u, 0 u 6 2p, to f...
 7.6.15: In 718, use the information given about the angle u, 0 u 6 2p, to f...
 7.6.16: In 718, use the information given about the angle u, 0 u 6 2p, to f...
 7.6.17: In 718, use the information given about the angle u, 0 u 6 2p, to f...
 7.6.18: In 718, use the information given about the angle u, 0 u 6 2p, to f...
 7.6.19: In 1928, use the Halfangle Formulas to find the exact value of eac...
 7.6.20: In 1928, use the Halfangle Formulas to find the exact value of eac...
 7.6.21: In 1928, use the Halfangle Formulas to find the exact value of eac...
 7.6.22: In 1928, use the Halfangle Formulas to find the exact value of eac...
 7.6.23: In 1928, use the Halfangle Formulas to find the exact value of eac...
 7.6.24: In 1928, use the Halfangle Formulas to find the exact value of eac...
 7.6.25: In 1928, use the Halfangle Formulas to find the exact value of eac...
 7.6.26: In 1928, use the Halfangle Formulas to find the exact value of eac...
 7.6.27: In 1928, use the Halfangle Formulas to find the exact value of eac...
 7.6.28: In 1928, use the Halfangle Formulas to find the exact value of eac...
 7.6.29: In 29 40, use the figures to evaluate each function given that f1x2...
 7.6.30: In 29 40, use the figures to evaluate each function given that f1x2...
 7.6.31: In 29 40, use the figures to evaluate each function given that f1x2...
 7.6.32: In 29 40, use the figures to evaluate each function given that f1x2...
 7.6.33: In 29 40, use the figures to evaluate each function given that f1x2...
 7.6.34: In 29 40, use the figures to evaluate each function given that f1x2...
 7.6.35: In 29 40, use the figures to evaluate each function given that f1x2...
 7.6.36: In 29 40, use the figures to evaluate each function given that f1x2...
 7.6.37: In 29 40, use the figures to evaluate each function given that f1x2...
 7.6.38: In 29 40, use the figures to evaluate each function given that f1x2...
 7.6.39: In 29 40, use the figures to evaluate each function given that f1x2...
 7.6.40: In 29 40, use the figures to evaluate each function given that f1x2...
 7.6.41: Show that sin4 u = 3 8  1 2 cos12u2 + 1 8 cos14u2
 7.6.42: Develop a formula for cos13u2 as a thirddegree polynomial in the v...
 7.6.43: Find an expression for sin15u2 as a fifthdegree polynomial in the ...
 7.6.44: Show that sin14u2 = 1cos u2 14 sin u  8 sin3 u2.
 7.6.45: Develop a formula for cos14u2 as a fourthdegree polynomial in the ...
 7.6.46: Find an expression for cos15u2 as a fifthdegree polynomial in the ...
 7.6.47: In 47 68, establish each identity.
 7.6.48: In 47 68, establish each identity.
 7.6.49: In 47 68, establish each identity.
 7.6.50: In 47 68, establish each identity.
 7.6.51: In 47 68, establish each identity.
 7.6.52: In 47 68, establish each identity.
 7.6.53: In 47 68, establish each identity.
 7.6.54: In 47 68, establish each identity.
 7.6.55: In 47 68, establish each identity.
 7.6.56: In 47 68, establish each identity.
 7.6.57: In 47 68, establish each identity.
 7.6.58: In 47 68, establish each identity.
 7.6.59: In 47 68, establish each identity.
 7.6.60: In 47 68, establish each identity.
 7.6.61: In 47 68, establish each identity.
 7.6.62: In 47 68, establish each identity.
 7.6.63: In 47 68, establish each identity.
 7.6.64: In 47 68, establish each identity.
 7.6.65: In 47 68, establish each identity.
 7.6.66: In 47 68, establish each identity.
 7.6.67: In 47 68, establish each identity.
 7.6.68: In 47 68, establish each identity.
 7.6.69: In 6978, solve each equation on the interval 0 u 6 2p.
 7.6.70: In 6978, solve each equation on the interval 0 u 6 2p.
 7.6.71: In 6978, solve each equation on the interval 0 u 6 2p.
 7.6.72: In 6978, solve each equation on the interval 0 u 6 2p.
 7.6.73: In 6978, solve each equation on the interval 0 u 6 2p.
 7.6.74: In 6978, solve each equation on the interval 0 u 6 2p.
 7.6.75: In 6978, solve each equation on the interval 0 u 6 2p.
 7.6.76: In 6978, solve each equation on the interval 0 u 6 2p.
 7.6.77: In 6978, solve each equation on the interval 0 u 6 2p.
 7.6.78: In 6978, solve each equation on the interval 0 u 6 2p.
 7.6.79: In 7990, find the exact value of each expression.
 7.6.80: In 7990, find the exact value of each expression.
 7.6.81: In 7990, find the exact value of each expression.
 7.6.82: In 7990, find the exact value of each expression.
 7.6.83: In 7990, find the exact value of each expression.
 7.6.84: In 7990, find the exact value of each expression.
 7.6.85: In 7990, find the exact value of each expression.
 7.6.86: In 7990, find the exact value of each expression.
 7.6.87: In 7990, find the exact value of each expression.
 7.6.88: In 7990, find the exact value of each expression.
 7.6.89: In 7990, find the exact value of each expression.
 7.6.90: In 7990, find the exact value of each expression.
 7.6.91: In 9193, find the real zeros of each trigonometric function on the ...
 7.6.92: In 9193, find the real zeros of each trigonometric function on the ...
 7.6.93: In 9193, find the real zeros of each trigonometric function on the ...
 7.6.94: A rain gutter is to be constructed of aluminum sheets 12 inches wid...
 7.6.95: In a laser projection system, the optical or scanning angle u is re...
 7.6.96: The product of inertia for an area about inclined axes is given by ...
 7.6.97: An object is propelled upward at an angle u, 45 6 u 6 90, to the ho...
 7.6.98: An oscilloscope often displays a sawtooth curve. This curve can be ...
 7.6.99: Show that the area A of an isosceles triangle whose equal sides are...
 7.6.100: A rectangle is inscribed in a semicircle of radius 1. See the illus...
 7.6.101: If x = 2 tan u, express sin12u2 as a function of x.
 7.6.102: If x = 2 tan u, express cos12u2 as a function of x
 7.6.103: Find the value of the number C: 1 2 sin2 x + C =  1 4 cos12x2
 7.6.104: Find the value of the number C: 1 2 cos2 x + C = 1 4 cos12x2
 7.6.105: If z = tan a 2 , show that sin a = 2z 1 + z2
 7.6.106: If z = tan a 2 , show that cos a = 1  z2 1 + z2 .
 7.6.107: Graph f1x2 = sin2 x = 1  cos12x2 2 for 0 x 2p by hand using transf...
 7.6.108: Repeat for g1x2 = cos2 x.
 7.6.109: Use the fact that cos p 12 = 1 4 1 26 + 222 to find sin p 24 and co...
 7.6.110: Show that cos p 8 = 32 + 22 2 and use it to find sin p 16 and cos p 16
 7.6.111: Show that sin3 u + sin3 1u + 1202 + sin3 1u + 2402 =  3 4 sin13u2
 7.6.112: If tan u = a tan u 3 , express tan u 3 in terms of a.
 7.6.113: Go to the library and research Chebyshv polynomials. Write a report...
Solutions for Chapter 7.6: Doubleangle and Halfangle Formulas
Full solutions for Precalculus Enhanced with Graphing Utilities  6th Edition
ISBN: 9780132854351
Solutions for Chapter 7.6: Doubleangle and Halfangle Formulas
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Precalculus Enhanced with Graphing Utilities was written by and is associated to the ISBN: 9780132854351. This textbook survival guide was created for the textbook: Precalculus Enhanced with Graphing Utilities, edition: 6. Since 113 problems in chapter 7.6: Doubleangle and Halfangle Formulas have been answered, more than 54371 students have viewed full stepbystep solutions from this chapter. Chapter 7.6: Doubleangle and Halfangle Formulas includes 113 full stepbystep solutions.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

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

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

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

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).

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

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

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

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Outer product uv T
= column times row = rank one matrix.

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

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.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

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

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

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.