 1.1.1: One degree, written 1, represents of a complete rotation.
 1.1.2: If the measure of an angle is x, its complement can be expressed as...
 1.1.3: If the measure of an angle is x, its supplement can be expressed as...
 1.1.4: The measure of an angle that is its own complement is .
 1.1.5: The measure of an angle that is its own supplement is .
 1.1.6: One minute, written 1, is of a degree.
 1.1.7: One second, written 1, is of a minute.
 1.1.8: 12 30 written in decimal degrees is .
 1.1.9: 55.25 written in degrees and minutes is .
 1.1.10: If n represents any integer, then an expression representing all an...
 1.1.11: Find the measure of (a) the complement and (b) the supplement of an...
 1.1.12: Find the measure of (a) the complement and (b) the supplement of an...
 1.1.13: Find the measure of (a) the complement and (b) the supplement of an...
 1.1.14: Find the measure of (a) the complement and (b) the supplement of an...
 1.1.15: Find the measure of (a) the complement and (b) the supplement of an...
 1.1.16: Find the measure of (a) the complement and (b) the supplement of an...
 1.1.17: Find the measure of (a) the complement and (b) the supplement of an...
 1.1.18: Find the measure of (a) the complement and (b) the supplement of an...
 1.1.19: Find the measure of (a) the complement and (b) the supplement of an...
 1.1.20: Find the measure of (a) the complement and (b) the supplement of an...
 1.1.21: Find the measure of (a) the complement and (b) the supplement of an...
 1.1.22: Find the measure of (a) the complement and (b) the supplement of an...
 1.1.23: Find the measure of each marked angle. See Example 2.
 1.1.24: Find the measure of each marked angle. See Example 2.
 1.1.25: Find the measure of each marked angle. See Example 2.
 1.1.26: Find the measure of each marked angle. See Example 2.
 1.1.27: Find the measure of each marked angle. See Example 2.
 1.1.28: Find the measure of each marked angle. See Example 2.
 1.1.29: 29. supplementary angles with measures 10x + 7 and 7x + 3 degrees
 1.1.30: 30. supplementary angles with measures 6x  4 and 8x  12 degrees
 1.1.31: 31. complementary angles with measures 9x + 6 and 3x degrees
 1.1.32: 32. complementary angles with measures 3x  5 and 6x  40 degrees
 1.1.33: Find the measure of the smaller angle formed by the hands of a cloc...
 1.1.34: Find the measure of the smaller angle formed by the hands of a cloc...
 1.1.35: Find the measure of the smaller angle formed by the hands of a cloc...
 1.1.36: Find the measure of the smaller angle formed by the hands of a cloc...
 1.1.37: Find the measure of the smaller angle formed by the hands of a cloc...
 1.1.38: Find the measure of the smaller angle formed by the hands of a cloc...
 1.1.39: Perform each calculation. See Example 3. 62 18 + 21 41
 1.1.40: Perform each calculation. See Example 3. 75 15 + 83 32
 1.1.41: Perform each calculation. See Example 3. 97 42 + 81 37
 1.1.42: Perform each calculation. See Example 3. 110 25 + 32 55
 1.1.43: Perform each calculation. See Example 3. 47 29  71 18
 1.1.44: Perform each calculation. See Example 3. 47 23  73 48
 1.1.45: Perform each calculation. See Example 3. 90  51 28
 1.1.46: Perform each calculation. See Example 3. 90  17 13
 1.1.47: Perform each calculation. See Example 3. 180  119 26
 1.1.48: Perform each calculation. See Example 3. 180  124 51
 1.1.49: Perform each calculation. See Example 3. 90  72 58 11
 1.1.50: Perform each calculation. See Example 3. 90  36 18 47
 1.1.51: Perform each calculation. See Example 3. 26 20 + 18 17  14 10
 1.1.52: Perform each calculation. See Example 3. 55 30 + 12 44  8 15
 1.1.53: Convert each angle measure to decimal degrees. If applicable, round...
 1.1.54: Convert each angle measure to decimal degrees. If applicable, round...
 1.1.55: Convert each angle measure to decimal degrees. If applicable, round...
 1.1.56: Convert each angle measure to decimal degrees. If applicable, round...
 1.1.57: Convert each angle measure to decimal degrees. If applicable, round...
 1.1.58: Convert each angle measure to decimal degrees. If applicable, round...
 1.1.59: Convert each angle measure to decimal degrees. If applicable, round...
 1.1.60: Convert each angle measure to decimal degrees. If applicable, round...
 1.1.61: Convert each angle measure to decimal degrees. If applicable, round...
 1.1.62: Convert each angle measure to decimal degrees. If applicable, round...
 1.1.63: Convert each angle measure to decimal degrees. If applicable, round...
 1.1.64: Convert each angle measure to decimal degrees. If applicable, round...
 1.1.65: Convert each angle measure to degrees, minutes, and seconds. If app...
 1.1.66: Convert each angle measure to degrees, minutes, and seconds. If app...
 1.1.67: Convert each angle measure to degrees, minutes, and seconds. If app...
 1.1.68: Convert each angle measure to degrees, minutes, and seconds. If app...
 1.1.69: Convert each angle measure to degrees, minutes, and seconds. If app...
 1.1.70: Convert each angle measure to degrees, minutes, and seconds. If app...
 1.1.71: Convert each angle measure to degrees, minutes, and seconds. If app...
 1.1.72: Convert each angle measure to degrees, minutes, and seconds. If app...
 1.1.73: Convert each angle measure to degrees, minutes, and seconds. If app...
 1.1.74: Convert each angle measure to degrees, minutes, and seconds. If app...
 1.1.75: Convert each angle measure to degrees, minutes, and seconds. If app...
 1.1.76: Convert each angle measure to degrees, minutes, and seconds. If app...
 1.1.77: Find the angle of least positive measure (not equal to the given me...
 1.1.78: Find the angle of least positive measure (not equal to the given me...
 1.1.79: Find the angle of least positive measure (not equal to the given me...
 1.1.80: Find the angle of least positive measure (not equal to the given me...
 1.1.81: Find the angle of least positive measure (not equal to the given me...
 1.1.82: Find the angle of least positive measure (not equal to the given me...
 1.1.83: Find the angle of least positive measure (not equal to the given me...
 1.1.84: Find the angle of least positive measure (not equal to the given me...
 1.1.85: Find the angle of least positive measure (not equal to the given me...
 1.1.86: Find the angle of least positive measure (not equal to the given me...
 1.1.87: Find the angle of least positive measure (not equal to the given me...
 1.1.88: Find the angle of least positive measure (not equal to the given me...
 1.1.89: Find the angle of least positive measure (not equal to the given me...
 1.1.90: Find the angle of least positive measure (not equal to the given me...
 1.1.91: Find the angle of least positive measure (not equal to the given me...
 1.1.92: Find the angle of least positive measure (not equal to the given me...
 1.1.93: Find the angle of least positive measure (not equal to the given me...
 1.1.94: Find the angle of least positive measure (not equal to the given me...
 1.1.95: Find the angle of least positive measure (not equal to the given me...
 1.1.96: Find the angle of least positive measure (not equal to the given me...
 1.1.97: Give two positive and two negative angles that are coterminal with ...
 1.1.98: Give two positive and two negative angles that are coterminal with ...
 1.1.99: Give two positive and two negative angles that are coterminal with ...
 1.1.100: Give two positive and two negative angles that are coterminal with ...
 1.1.101: Write an expression that generates all angles coterminal with each ...
 1.1.102: Write an expression that generates all angles coterminal with each ...
 1.1.103: Write an expression that generates all angles coterminal with each ...
 1.1.104: Write an expression that generates all angles coterminal with each ...
 1.1.105: Write an expression that generates all angles coterminal with each ...
 1.1.106: Write an expression that generates all angles coterminal with each ...
 1.1.107: Write an expression that generates all angles coterminal with each ...
 1.1.108: Write an expression that generates all angles coterminal with each ...
 1.1.109: Why do the answers to Exercises 107 and 108 give the same set of an...
 1.1.110: Concept Check Which two of the following are not coterminal with r?...
 1.1.111: Concept Check Sketch each angle in standard position. Draw an arrow...
 1.1.112: Concept Check Sketch each angle in standard position. Draw an arrow...
 1.1.113: Concept Check Sketch each angle in standard position. Draw an arrow...
 1.1.114: Concept Check Sketch each angle in standard position. Draw an arrow...
 1.1.115: Concept Check Sketch each angle in standard position. Draw an arrow...
 1.1.116: Concept Check Sketch each angle in standard position. Draw an arrow...
 1.1.117: Concept Check Sketch each angle in standard position. Draw an arrow...
 1.1.118: Concept Check Sketch each angle in standard position. Draw an arrow...
 1.1.119: Concept Check Sketch each angle in standard position. Draw an arrow...
 1.1.120: Concept Check Sketch each angle in standard position. Draw an arrow...
 1.1.121: Concept Check Sketch each angle in standard position. Draw an arrow...
 1.1.122: Concept Check Sketch each angle in standard position. Draw an arrow...
 1.1.123: Solve each problem. See Example 6. Revolutions of a Turntable A tur...
 1.1.124: Solve each problem. See Example 6. Revolutions of a Windmill A wind...
 1.1.125: Solve each problem. See Example 6. Rotating Tire A tire is rotating...
 1.1.126: Rotating Airplane Propeller An airplane propeller rotates 1000 time...
 1.1.127: Rotating Pulley A pulley rotates through 75 in 1 min. How many rota...
 1.1.128: Surveying One student in a surveying class measures an angle as 74....
 1.1.129: Viewing Field of a Telescope As a consequence of Earths rotation, c...
 1.1.130: Angle Measure of a Star on the American Flag Determine the measure ...
Solutions for Chapter 1.1: Angles
Full solutions for Trigonometry  11th Edition
ISBN: 9780134217437
Solutions for Chapter 1.1: Angles
Get Full SolutionsThis textbook survival guide was created for the textbook: Trigonometry, edition: 11. Chapter 1.1: Angles includes 130 full stepbystep solutions. Trigonometry was written by and is associated to the ISBN: 9780134217437. Since 130 problems in chapter 1.1: Angles have been answered, more than 25949 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions.

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.)

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

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

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.

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

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.

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.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)ยท(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).

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

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

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

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

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

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

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).