 5.5.1: Match each expression in Column I with its value in Column II. 2 co...
 5.5.2: Match each expression in Column I with its value in Column II. 2 ta...
 5.5.3: Match each expression in Column I with its value in Column II. 2 si...
 5.5.4: Match each expression in Column I with its value in Column II. cos2...
 5.5.5: Match each expression in Column I with its value in Column II. 4 si...
 5.5.6: Match each expression in Column I with its value in Column II. 2 ta...
 5.5.7: Find values of the sine and cosine functions for each angle measure...
 5.5.8: Find values of the sine and cosine functions for each angle measure...
 5.5.9: Find values of the sine and cosine functions for each angle measure...
 5.5.10: Find values of the sine and cosine functions for each angle measure...
 5.5.11: Find values of the sine and cosine functions for each angle measure...
 5.5.12: Find values of the sine and cosine functions for each angle measure...
 5.5.13: Find values of the sine and cosine functions for each angle measure...
 5.5.14: Find values of the sine and cosine functions for each angle measure...
 5.5.15: Find values of the sine and cosine functions for each angle measure...
 5.5.16: Find values of the sine and cosine functions for each angle measure...
 5.5.17: Verify that each equation is an identity. See Example 3. 17. 1sin x...
 5.5.18: Verify that each equation is an identity. See Example 3. 18. sec 2x...
 5.5.19: Verify that each equation is an identity. See Example 3. 19. 1cos 2...
 5.5.20: Verify that each equation is an identity. See Example 3. 20. 1cos 2...
 5.5.21: Verify that each equation is an identity. See Example 3. 21. tan 8u...
 5.5.22: Verify that each equation is an identity. See Example 3. 22. sin 2x...
 5.5.23: Verify that each equation is an identity. See Example 3. 23. cos 2u...
 5.5.24: Verify that each equation is an identity. See Example 3. 24. tan 2u...
 5.5.25: Verify that each equation is an identity. See Example 3. 25. sin 4x...
 5.5.26: Verify that each equation is an identity. See Example 3. 26. 1 + co...
 5.5.27: Verify that each equation is an identity. See Example 3. 27. 2 cos ...
 5.5.28: Verify that each equation is an identity. See Example 3. 28. cot 4u...
 5.5.29: Verify that each equation is an identity. See Example 3. 29. tan x ...
 5.5.30: Verify that each equation is an identity. See Example 3. 30. cos 2x...
 5.5.31: Verify that each equation is an identity. See Example 3. 31. 1 + ta...
 5.5.32: Verify that each equation is an identity. See Example 3. 32. cot A ...
 5.5.33: Verify that each equation is an identity. See Example 3. 33. sin 2A...
 5.5.34: Verify that each equation is an identity. See Example 3. 34. sin 4x...
 5.5.35: Verify that each equation is an identity. See Example 3. 35. tan1u ...
 5.5.36: Verify that each equation is an identity. See Example 3. 36. cot u ...
 5.5.37: Simplify each expression. See Example 4. cos2 15  sin2 15
 5.5.38: Simplify each expression. See Example 4. 2 tan 15 1  tan2 15
 5.5.39: Simplify each expression. See Example 4. 1  2 sin2 15
 5.5.40: Simplify each expression. See Example 4. 1  2 sin2 22 1 2
 5.5.41: Simplify each expression. See Example 4. 2 cos2 67 1 2  1
 5.5.42: Simplify each expression. See Example 4. cos2 p 8  1 2
 5.5.43: Simplify each expression. See Example 4. tan 51 1  tan2 51
 5.5.44: Simplify each expression. See Example 4. tan 34 211  tan2 342
 5.5.45: Simplify each expression. See Example 4. 1 4  1 2 sin2 47.1
 5.5.46: Simplify each expression. See Example 4. 1 8 sin 29.5 cos 29.5
 5.5.47: Simplify each expression. See Example 4. sin2 2p 5  cos2 2p 5
 5.5.48: Simplify each expression. See Example 4. cos2 2x  sin2 2x
 5.5.49: Express each function as a trigonometric function of x. See Example...
 5.5.50: Express each function as a trigonometric function of x. See Example...
 5.5.51: Express each function as a trigonometric function of x. See Example...
 5.5.52: Express each function as a trigonometric function of x. See Example...
 5.5.53: Graph each expression and use the graph to make a conjecture, predi...
 5.5.54: Graph each expression and use the graph to make a conjecture, predi...
 5.5.55: Graph each expression and use the graph to make a conjecture, predi...
 5.5.56: Graph each expression and use the graph to make a conjecture, predi...
 5.5.57: Write each expression as a sum or difference of trigonometric funct...
 5.5.58: Write each expression as a sum or difference of trigonometric funct...
 5.5.59: Write each expression as a sum or difference of trigonometric funct...
 5.5.60: Write each expression as a sum or difference of trigonometric funct...
 5.5.61: Write each expression as a sum or difference of trigonometric funct...
 5.5.62: Write each expression as a sum or difference of trigonometric funct...
 5.5.63: Write each expression as a product of trigonometric functions. See ...
 5.5.64: Write each expression as a product of trigonometric functions. See ...
 5.5.65: Write each expression as a product of trigonometric functions. See ...
 5.5.66: Write each expression as a product of trigonometric functions. See ...
 5.5.67: Write each expression as a product of trigonometric functions. See ...
 5.5.68: Write each expression as a product of trigonometric functions. See ...
 5.5.69: Solve each problem. See Example 6. Wattage Consumption Use the iden...
 5.5.70: Solve each problem. See Example 6. Amperage, Wattage, and Voltage A...
Solutions for Chapter 5.5: DoubleAngle Identities
Full solutions for Trigonometry  11th Edition
ISBN: 9780134217437
Solutions for Chapter 5.5: DoubleAngle Identities
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Trigonometry, edition: 11. Chapter 5.5: DoubleAngle Identities includes 70 full stepbystep solutions. Since 70 problems in chapter 5.5: DoubleAngle Identities have been answered, more than 25873 students have viewed full stepbystep solutions from this chapter. Trigonometry was written by and is associated to the ISBN: 9780134217437.

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.

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

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).

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.

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

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

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.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

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

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.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

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.

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!

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.

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.

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