 6.1: Graph the inverse sine, cosine, and tangent functions, indicating t...
 6.2: The ranges of the inverse tangent and inverse cotangent functions a...
 6.3: It is true that sin 11p6 =  12, and therefore arcsin A  12B = 11p6 .
 6.4: For all x, tan1tan1 x2 = x.
 6.5: Find the exact value of each real number y. Do not use a calculator...
 6.6: Find the exact value of each real number y. Do not use a calculator...
 6.7: Find the exact value of each real number y. Do not use a calculator...
 6.8: Find the exact value of each real number y. Do not use a calculator...
 6.9: Find the exact value of each real number y. Do not use a calculator...
 6.10: Find the exact value of each real number y. Do not use a calculator...
 6.11: Find the exact value of each real number y. Do not use a calculator...
 6.12: Find the exact value of each real number y. Do not use a calculator...
 6.13: Find the exact value of each real number y. Do not use a calculator...
 6.14: Give the degree measure of u. Do not use a calculator. u = arccos12
 6.15: Give the degree measure of u. Do not use a calculator. u = arcsin 232
 6.16: Give the degree measure of u. Do not use a calculator. u = tan1 0
 6.17: Use a calculator to approximate each value in decimal degrees u = a...
 6.18: Use a calculator to approximate each value in decimal degrees u = s...
 6.19: Use a calculator to approximate each value in decimal degrees u = c...
 6.20: Use a calculator to approximate each value in decimal degrees u = c...
 6.21: Use a calculator to approximate each value in decimal degrees u = a...
 6.22: Use a calculator to approximate each value in decimal degrees u = c...
 6.23: Evaluate each expression without using a calculator. cos1arccos1122
 6.24: Evaluate each expression without using a calculator. sin arcsin 2322
 6.25: Evaluate each expression without using a calculator.arccos acos3p4b
 6.26: Evaluate each expression without using a calculator.arcsec1sec p2
 6.27: Evaluate each expression without using a calculator.tan1 atanp4b
 6.28: Evaluate each expression without using a calculator.cos11cos 02
 6.29: Evaluate each expression without using a calculator.sin aarccos34b
 6.30: Evaluate each expression without using a calculator.cos1arctan 32
 6.31: Evaluate each expression without using a calculator.cos1csc1122
 6.32: Evaluate each expression without using a calculator.sec a2 sin1 a...
 6.33: Evaluate each expression without using a calculator.tan aarcsin35 +...
 6.34: Write each trigonometric expression as an algebraic expression in u...
 6.35: Write each trigonometric expression as an algebraic expression in u...
 6.36: Solve each equation for exact solutions over the interval 30, 2p2 w...
 6.37: Solve each equation for exact solutions over the interval 30, 2p2 w...
 6.38: Solve each equation for exact solutions over the interval 30, 2p2 w...
 6.39: Solve each equation for exact solutions over the interval 30, 2p2 w...
 6.40: Solve each equation for exact solutions over the interval 30, 2p2 w...
 6.41: Solve each equation for exact solutions over the interval 30, 2p2 w...
 6.42: Solve each equation for all exact solutions, in radians. secx2 = cosx2
 6.43: Solve each equation for all exact solutions, in radians. cos 2x + c...
 6.44: Solve each equation for all exact solutions, in radians. 4 sin x co...
 6.45: Solve each equation for exact solutions over the interval 30 3602 w...
 6.46: Solve each equation for exact solutions over the interval 30 3602 w...
 6.47: Solve each equation for exact solutions over the interval 30 3602 w...
 6.48: Solve each equation for exact solutions over the interval 30 3602 w...
 6.49: Solve each equation for exact solutions over the interval 30 3602 w...
 6.50: Solve each equation for exact solutions over the interval 30 3602 w...
 6.51: Solve each equation for all exact solutions, in degrees. 223 cosu2 ...
 6.52: Solve each equation for all exact solutions, in degrees. sin u  co...
 6.53: Solve each equation for all exact solutions, in degrees. tan u  se...
 6.54: Solve each equation for x. 4p  4 cot1 x = p
 6.55: Solve each equation for x. 43arctanx2 = p
 6.56: Solve each equation for x. arccos x = arcsin27
 6.57: Solve each equation for x. arccos x + arctan 1 =11p12
 6.58: Solve each equation for x. y = 3 cosx2, for x in 30, 2p4
 6.59: Solve each equation for x. y =12sin x, for x in c p2,p2d
 6.60: Solve each equation for x. y =45sin x 35, for x in c p2,p2d
 6.61: Solve each equation for x. y =12tan13x + 22, for x in a23  p6, 2...
 6.62: Solve each equation for x. Solve d = 550 + 450 cos a p50tb for t, w...
 6.63: Viewing Angle of an Observer A 10ftwide chalkboardis situated 5 f...
 6.64: Snells Law Snells law states thatc1c2=sin u1sin u2,where c1 is the ...
 6.65: Snells Law Refer to Exercise 64. What happens when u1 is greater th...
 6.66: British Nautical Mile The British nauticalmile is defined as the le...
Solutions for Chapter 6: Inverse Circular Functions and Trigonometric Equations
Full solutions for Trigonometry  11th Edition
ISBN: 9780134217437
Solutions for Chapter 6: Inverse Circular Functions and Trigonometric Equations
Get Full SolutionsThis textbook survival guide was created for the textbook: Trigonometry, edition: 11. Since 66 problems in chapter 6: Inverse Circular Functions and Trigonometric Equations have been answered, more than 19511 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Trigonometry was written by and is associated to the ISBN: 9780134217437. Chapter 6: Inverse Circular Functions and Trigonometric Equations includes 66 full stepbystep solutions.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

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 II· Condition numbers measure the sensitivity of the output to change in the input.

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

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.

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.

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

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.

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

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

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

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.

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.