 7.5.1: What is the equation of the unit circle?
 7.5.2: The domain of the function is (pp. 200208) f1x2 = . 3x  6 x  4
 7.5.3: A function for which for all x in the domain of f is called a(n) fu...
 7.5.4: The sine, cosine, cosecant, and secant functions have period ; the ...
 7.5.5: Let t be a real number and let be the point on the unit circle that...
 7.5.6: For any angle in standard position, let be any point on the termina...
 7.5.7: If sin u = 0.2 , then sin1u2 = and sin1u + 2p2 =
 7.5.8: True or False The only even trigonometric functions are the cosine ...
 7.5.9: In 914, the point P on the unit circle that corresponds to a real n...
 7.5.10: In 914, the point P on the unit circle that corresponds to a real n...
 7.5.11: In 914, the point P on the unit circle that corresponds to a real n...
 7.5.12: In 914, the point P on the unit circle that corresponds to a real n...
 7.5.13: In 914, the point P on the unit circle that corresponds to a real n...
 7.5.14: In 914, the point P on the unit circle that corresponds to a real n...
 7.5.15: In 1520, the point P on the circle that is also on the terminal sid...
 7.5.16: In 1520, the point P on the circle that is also on the terminal sid...
 7.5.17: In 1520, the point P on the circle that is also on the terminal sid...
 7.5.18: In 1520, the point P on the circle that is also on the terminal sid...
 7.5.19: In 1520, the point P on the circle that is also on the terminal sid...
 7.5.20: In 1520, the point P on the circle that is also on the terminal sid...
 7.5.21: In 2136, use the fact that the trigonometric functions are periodic...
 7.5.22: In 2136, use the fact that the trigonometric functions are periodic...
 7.5.23: In 2136, use the fact that the trigonometric functions are periodic...
 7.5.24: In 2136, use the fact that the trigonometric functions are periodic...
 7.5.25: In 2136, use the fact that the trigonometric functions are periodic...
 7.5.26: In 2136, use the fact that the trigonometric functions are periodic...
 7.5.27: In 2136, use the fact that the trigonometric functions are periodic...
 7.5.28: In 2136, use the fact that the trigonometric functions are periodic...
 7.5.29: In 2136, use the fact that the trigonometric functions are periodic...
 7.5.30: In 2136, use the fact that the trigonometric functions are periodic...
 7.5.31: In 2136, use the fact that the trigonometric functions are periodic...
 7.5.32: In 2136, use the fact that the trigonometric functions are periodic...
 7.5.33: In 2136, use the fact that the trigonometric functions are periodic...
 7.5.34: In 2136, use the fact that the trigonometric functions are periodic...
 7.5.35: In 2136, use the fact that the trigonometric functions are periodic...
 7.5.36: In 2136, use the fact that the trigonometric functions are periodic...
 7.5.37: In 3754, use the evenodd properties to find the exact value of each...
 7.5.38: In 3754, use the evenodd properties to find the exact value of each...
 7.5.39: In 3754, use the evenodd properties to find the exact value of each...
 7.5.40: In 3754, use the evenodd properties to find the exact value of each...
 7.5.41: In 3754, use the evenodd properties to find the exact value of each...
 7.5.42: In 3754, use the evenodd properties to find the exact value of each...
 7.5.43: In 3754, use the evenodd properties to find the exact value of each...
 7.5.44: In 3754, use the evenodd properties to find the exact value of each...
 7.5.45: In 3754, use the evenodd properties to find the exact value of each...
 7.5.46: In 3754, use the evenodd properties to find the exact value of each...
 7.5.47: In 3754, use the evenodd properties to find the exact value of each...
 7.5.48: In 3754, use the evenodd properties to find the exact value of each...
 7.5.49: In 3754, use the evenodd properties to find the exact value of each...
 7.5.50: In 3754, use the evenodd properties to find the exact value of each...
 7.5.51: In 3754, use the evenodd properties to find the exact value of each...
 7.5.52: In 3754, use the evenodd properties to find the exact value of each...
 7.5.53: In 3754, use the evenodd properties to find the exact value of each...
 7.5.54: In 3754, use the evenodd properties to find the exact value of each...
 7.5.55: In 5560, find the exact value of each expression. Do not use a calc...
 7.5.56: In 5560, find the exact value of each expression. Do not use a calc...
 7.5.57: In 5560, find the exact value of each expression. Do not use a calc...
 7.5.58: In 5560, find the exact value of each expression. Do not use a calc...
 7.5.59: In 5560, find the exact value of each expression. Do not use a calc...
 7.5.60: In 5560, find the exact value of each expression. Do not use a calc...
 7.5.61: What is the domain of the sine function?
 7.5.62: What is the domain of the cosine function?
 7.5.63: For what numbers u is f1u2 = tan u not defined?
 7.5.64: For what numbers u is f1u2 = cot u not defined?
 7.5.65: For what numbers u is f1u2 = sec u not defined?
 7.5.66: For what numbers u is f1u2 = csc u not defined?
 7.5.67: What is the range of the sine function?
 7.5.68: What is the range of the cosine function?
 7.5.69: What is the range of the tangent function?
 7.5.70: What is the range of the cotangent function?
 7.5.71: What is the range of the secant function?
 7.5.72: What is the range of the cosecant function?
 7.5.73: Is the sine function even, odd, or neither? Is its graphsymmetric? ...
 7.5.74: Is the cosine function even, odd, or neither? Is its graph symmetri...
 7.5.75: Is the tangent function even, odd, or neither? Is its graph symmetr...
 7.5.76: Is the cotangent function even, odd, or neither? Is its graph symme...
 7.5.77: Is the secant function even, odd, or neither? Is its graph symmetri...
 7.5.78: Is the cosecant function even, odd, or neither? Is its graph symmet...
 7.5.79: If sin u = 0.3, find the value ofsin u + sin1u + 2p2 + sin1u + 4p2
 7.5.80: If cos u = 0.2, find the value ofcos u + cos1u + 2p2 + cos1u + 4p2
 7.5.81: If tan u = 3 , find the value oftan u + tan1u + p2 + tan1u + 2p2
 7.5.82: If cot u = 2, find the value ofcot u + cot1u  p2 + cot1u  2p2
 7.5.83: In 8388, use the periodic and evenodd properties.f1a2 f1a2 + f1a +...
 7.5.84: In 8388, use the periodic and evenodd properties.(a) f1a2 (b) f1a2...
 7.5.85: In 8388, use the periodic and evenodd properties.f1a2 f1a2 + f1a +...
 7.5.86: In 8388, use the periodic and evenodd properties.f1a2 (b) f1a2 + f...
 7.5.87: In 8388, use the periodic and evenodd properties.f1a2 f1a2 + f1a +...
 7.5.88: In 8388, use the periodic and evenodd properties.(a) f1a2 (b) f1a2...
 7.5.89: In 8990, use the figure to approximate the value of the six trigono...
 7.5.90: In 8990, use the figure to approximate the value of the six trigono...
 7.5.91: Show that the range of the tangent function is the set of allreal n...
 7.5.92: Show that the range of the cotangent function is the set ofall real...
 7.5.93: Show that the period of is . [Hint: Assume that exists so that for ...
 7.5.94: Show that the period of isf1u2 = cos u 2p
 7.5.95: Show that the period of isf1u2 = sec u 2p
 7.5.96: Show that the period of isf1u2 = csc u 2p
 7.5.97: Show that the period of isf1u2 = tan u p
 7.5.98: Show that the period of isf1u2 = cot u p
 7.5.99: If , ,isthe angle between a horizontalray directed to the right (sa...
 7.5.100: Explain how you would find the value of using periodic properties.
 7.5.101: Explain how you would find the value of cos using evenodd properties.
 7.5.102: Write down five properties of the tangent function. Explain the mea...
 7.5.103: Describe your understanding of the meaning of a periodic function.
Solutions for Chapter 7.5: Algebra and Trigonometry 9th Edition
Full solutions for Algebra and Trigonometry  9th Edition
ISBN: 9780321716569
Solutions for Chapter 7.5
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Since 103 problems in chapter 7.5 have been answered, more than 55167 students have viewed full stepbystep solutions from this chapter. Chapter 7.5 includes 103 full stepbystep solutions. This textbook survival guide was created for the textbook: Algebra and Trigonometry, edition: 9. Algebra and Trigonometry was written by and is associated to the ISBN: 9780321716569.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

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).

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

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.

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

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.

Iterative method.
A sequence of steps intended to approach the desired solution.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

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.

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

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

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

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!

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

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·

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