 5.1.1: For each expression in Column I, choose the expression from Column ...
 5.1.2: For each expression in Column I, choose the expression from Column ...
 5.1.3: For each expression in Column I, choose the expression from Column ...
 5.1.4: For each expression in Column I, choose the expression from Column ...
 5.1.5: For each expression in Column I, choose the expression from Column ...
 5.1.6: Use identities to correctly complete each sentence. If tan u = 2.6,...
 5.1.7: Use identities to correctly complete each sentence. If cos u = 0.6...
 5.1.8: Use identities to correctly complete each sentence. If tan u = 1.6,...
 5.1.9: Use identities to correctly complete each sentence. If cos u = 0.8 ...
 5.1.10: Use identities to correctly complete each sentence. If sin u = 23 ,...
 5.1.11: Find sin u. See Example 1. cos u = 34 , u in quadrant I
 5.1.12: Find sin u. See Example 1. cos u = 56 , u in quadrant I
 5.1.13: Find sin u. See Example 1. cot u =  15 , u in quadrant IV
 5.1.14: Find sin u. See Example 1. cot u =  13 , u in quadrant IV
 5.1.15: Find sin u. See Example 1. cos1u2 = 255 , tan u 6 0
 5.1.16: Find sin u. See Example 1. cos1u2 = 236 , cot u 6 0
 5.1.17: Find sin u. See Example 1. tan u =  262 , cos u 7 0
 5.1.18: Find sin u. See Example 1. tan u =  272 , sec u 7 0
 5.1.19: Find sin u. See Example 1. sec u = 114 , cot u 6 0
 5.1.20: Find sin u. See Example 1. sec u = 72 , tan u 6 0
 5.1.21: Find sin u. See Example 1. csc u =  94
 5.1.22: Find sin u. See Example 1. csc u =  85
 5.1.23: Why is it unnecessary to give the quadrant of u in Exercises 21 and...
 5.1.24: Concept Check What is WRONG with the statement of this problem? Fin...
 5.1.25: 1x2 = sin x x
 5.1.26: 1x2 = x cos x
 5.1.27: Identify the basic trigonometric function graphed and determine whe...
 5.1.28: Identify the basic trigonometric function graphed and determine whe...
 5.1.29: Identify the basic trigonometric function graphed and determine whe...
 5.1.30: Identify the basic trigonometric function graphed and determine whe...
 5.1.31: Find the remaining five trigonometric functions of u. See Example 1...
 5.1.32: Find the remaining five trigonometric functions of u. See Example 1...
 5.1.33: Find the remaining five trigonometric functions of u. See Example 1...
 5.1.34: Find the remaining five trigonometric functions of u. See Example 1...
 5.1.35: Find the remaining five trigonometric functions of u. See Example 1...
 5.1.36: Find the remaining five trigonometric functions of u. See Example 1...
 5.1.37: Find the remaining five trigonometric functions of u. See Example 1...
 5.1.38: Find the remaining five trigonometric functions of u. See Example 1...
 5.1.39: For each expression in Column I, choose the expression from Column ...
 5.1.40: For each expression in Column I, choose the expression from Column ...
 5.1.41: For each expression in Column I, choose the expression from Column ...
 5.1.42: For each expression in Column I, choose the expression from Column ...
 5.1.43: For each expression in Column I, choose the expression from Column ...
 5.1.44: A student writes 1 + cot2 = csc2. Comment on this students work.
 5.1.45: Concept Check Suppose that cos u = xx + 1 . Find an expression in x...
 5.1.46: Concept Check Suppose that sec u = x + 4x . Find an expression in x...
 5.1.47: Perform each transformation. See Example 2. Write sin x in terms of...
 5.1.48: Perform each transformation. See Example 2. Write cot x in terms of...
 5.1.49: Perform each transformation. See Example 2. Write tan x in terms of...
 5.1.50: Perform each transformation. See Example 2. Write cot x in terms of...
 5.1.51: Perform each transformation. See Example 2. Write csc x in terms of...
 5.1.52: Perform each transformation. See Example 2. Write sec x in terms of...
 5.1.53: Write each expression in terms of sine and cosine, and then simplif...
 5.1.54: Write each expression in terms of sine and cosine, and then simplif...
 5.1.55: Write each expression in terms of sine and cosine, and then simplif...
 5.1.56: Write each expression in terms of sine and cosine, and then simplif...
 5.1.57: Write each expression in terms of sine and cosine, and then simplif...
 5.1.58: Write each expression in terms of sine and cosine, and then simplif...
 5.1.59: Write each expression in terms of sine and cosine, and then simplif...
 5.1.60: Write each expression in terms of sine and cosine, and then simplif...
 5.1.61: Write each expression in terms of sine and cosine, and then simplif...
 5.1.62: Write each expression in terms of sine and cosine, and then simplif...
 5.1.63: Write each expression in terms of sine and cosine, and then simplif...
 5.1.64: Write each expression in terms of sine and cosine, and then simplif...
 5.1.65: Write each expression in terms of sine and cosine, and then simplif...
 5.1.66: Write each expression in terms of sine and cosine, and then simplif...
 5.1.67: Write each expression in terms of sine and cosine, and then simplif...
 5.1.68: Write each expression in terms of sine and cosine, and then simplif...
 5.1.69: Write each expression in terms of sine and cosine, and then simplif...
 5.1.70: Write each expression in terms of sine and cosine, and then simplif...
 5.1.71: Write each expression in terms of sine and cosine, and then simplif...
 5.1.72: Write each expression in terms of sine and cosine, and then simplif...
 5.1.73: Write each expression in terms of sine and cosine, and then simplif...
 5.1.74: Write each expression in terms of sine and cosine, and then simplif...
 5.1.75: Write each expression in terms of sine and cosine, and then simplif...
 5.1.76: Write each expression in terms of sine and cosine, and then simplif...
 5.1.77: Write each expression in terms of sine and cosine, and then simplif...
 5.1.78: Write each expression in terms of sine and cosine, and then simplif...
 5.1.79: Work each problem. Let cos x = 15 . Find all possible values of sec...
 5.1.80: Work each problem. Let csc x = 3. Find all possible values of sin ...
 5.1.81: Use a graphing calculator to make a conjecture about whether each e...
 5.1.82: Use a graphing calculator to make a conjecture about whether each e...
 5.1.83: Use a graphing calculator to make a conjecture about whether each e...
 5.1.84: Use a graphing calculator to make a conjecture about whether each e...
 5.1.85: For individual or collaborative investigation (Exercises 8590) Prev...
 5.1.86: For individual or collaborative investigation (Exercises 8590) Prev...
 5.1.87: For individual or collaborative investigation (Exercises 8590) Prev...
 5.1.88: For individual or collaborative investigation (Exercises 8590) Prev...
 5.1.89: For individual or collaborative investigation (Exercises 8590) Prev...
 5.1.90: For individual or collaborative investigation (Exercises 8590) Prev...
Solutions for Chapter 5.1: Fundamental Identities
Full solutions for Trigonometry  11th Edition
ISBN: 9780134217437
Solutions for Chapter 5.1: Fundamental Identities
Get Full SolutionsChapter 5.1: Fundamental Identities includes 90 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 90 problems in chapter 5.1: Fundamental Identities have been answered, more than 25959 students have viewed full stepbystep solutions from this chapter. Trigonometry was written by and is associated to the ISBN: 9780134217437. This textbook survival guide was created for the textbook: Trigonometry, edition: 11.

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.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

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

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib IIĀ· Condition numbers measure the sensitivity of the output to change in the input.

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.

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.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

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

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

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

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

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.

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

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

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.

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.

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

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.