 3.3.1: For each value of the real number s, find (a) sin s , (b) cos s , a...
 3.3.2: For each value of the real number s, find (a) sin s , (b) cos s , a...
 3.3.3: For each value of the real number s, find (a) sin s , (b) cos s , a...
 3.3.4: For each value of the real number s, find (a) sin s , (b) cos s , a...
 3.3.5: For each value of the real number s, find (a) sin s , (b) cos s , a...
 3.3.6: For each value of the real number s, find (a) sin s , (b) cos s , a...
 3.3.7: Find the exact circular function value for each of the following. S...
 3.3.8: Find the exact circular function value for each of the following. S...
 3.3.9: Find the exact circular function value for each of the following. S...
 3.3.10: Find the exact circular function value for each of the following. S...
 3.3.11: Find the exact circular function value for each of the following. S...
 3.3.12: Find the exact circular function value for each of the following. S...
 3.3.13: Find the exact circular function value for each of the following. S...
 3.3.14: Find the exact circular function value for each of the following. S...
 3.3.15: Find the exact circular function value for each of the following. S...
 3.3.16: Find the exact circular function value for each of the following. S...
 3.3.17: Find the exact circular function value for each of the following. S...
 3.3.18: Find the exact circular function value for each of the following. S...
 3.3.19: Find the exact circular function value for each of the following. S...
 3.3.20: Find the exact circular function value for each of the following. S...
 3.3.21: Find the exact circular function value for each of the following. S...
 3.3.22: Find the exact circular function value for each of the following. S...
 3.3.23: Find a calculator approximation for each circular function value. S...
 3.3.24: Find a calculator approximation for each circular function value. S...
 3.3.25: Find a calculator approximation for each circular function value. S...
 3.3.26: Find a calculator approximation for each circular function value. S...
 3.3.27: Find a calculator approximation for each circular function value. S...
 3.3.28: Find a calculator approximation for each circular function value. S...
 3.3.29: Find a calculator approximation for each circular function value. S...
 3.3.30: Find a calculator approximation for each circular function value. S...
 3.3.31: Find a calculator approximation for each circular function value. S...
 3.3.32: Find a calculator approximation for each circular function value. S...
 3.3.33: Find a calculator approximation for each circular function value. S...
 3.3.34: Find a calculator approximation for each circular function value. S...
 3.3.35: Concept Check The figure displays a unit circle and an angle of 1 r...
 3.3.36: Concept Check The figure displays a unit circle and an angle of 1 r...
 3.3.37: Concept Check The figure displays a unit circle and an angle of 1 r...
 3.3.38: Concept Check The figure displays a unit circle and an angle of 1 r...
 3.3.39: Concept Check The figure displays a unit circle and an angle of 1 r...
 3.3.40: Concept Check The figure displays a unit circle and an angle of 1 r...
 3.3.41: Concept Check The figure displays a unit circle and an angle of 1 r...
 3.3.42: Concept Check The figure displays a unit circle and an angle of 1 r...
 3.3.43: Concept Check The figure displays a unit circle and an angle of 1 r...
 3.3.44: Concept Check The figure displays a unit circle and an angle of 1 r...
 3.3.45: Concept Check Without using a calculator, decide whether each funct...
 3.3.46: Concept Check Without using a calculator, decide whether each funct...
 3.3.47: Concept Check Without using a calculator, decide whether each funct...
 3.3.48: Concept Check Without using a calculator, decide whether each funct...
 3.3.49: Concept Check Without using a calculator, decide whether each funct...
 3.3.50: Concept Check Without using a calculator, decide whether each funct...
 3.3.51: Concept Check Each figure in Exercises 5154 shows an angle u in sta...
 3.3.52: Concept Check Each figure in Exercises 5154 shows an angle u in sta...
 3.3.53: Concept Check Each figure in Exercises 5154 shows an angle u in sta...
 3.3.54: Concept Check Each figure in Exercises 5154 shows an angle u in sta...
 3.3.55: Find the value of s in the interval 30, p2 4 that makes each statem...
 3.3.56: Find the value of s in the interval 30, p2 4 that makes each statem...
 3.3.57: Find the value of s in the interval 30, p2 4 that makes each statem...
 3.3.58: Find the value of s in the interval 30, p2 4 that makes each statem...
 3.3.59: Find the value of s in the interval 30, p2 4 that makes each statem...
 3.3.60: Find the value of s in the interval 30, p2 4 that makes each statem...
 3.3.61: Find the exact value of s in the given interval that has the given ...
 3.3.62: Find the exact value of s in the given interval that has the given ...
 3.3.63: Find the exact value of s in the given interval that has the given ...
 3.3.64: Find the exact value of s in the given interval that has the given ...
 3.3.65: Find the exact value of s in the given interval that has the given ...
 3.3.66: Find the exact value of s in the given interval that has the given ...
 3.3.67: Find the exact values of s in the given interval that satisfy the g...
 3.3.68: Find the exact values of s in the given interval that satisfy the g...
 3.3.69: Find the exact values of s in the given interval that satisfy the g...
 3.3.70: Find the exact values of s in the given interval that satisfy the g...
 3.3.71: Find the exact values of s in the given interval that satisfy the g...
 3.3.72: Find the exact values of s in the given interval that satisfy the g...
 3.3.73: Suppose an arc of length s lies on the unit circle x2 + y2 = 1 , st...
 3.3.74: Suppose an arc of length s lies on the unit circle x2 + y2 = 1 , st...
 3.3.75: Suppose an arc of length s lies on the unit circle x2 + y2 = 1 , st...
 3.3.76: Suppose an arc of length s lies on the unit circle x2 + y2 = 1 , st...
 3.3.77: Concept Check For each value of s, use a calculator to find sin s a...
 3.3.78: Concept Check For each value of s, use a calculator to find sin s a...
 3.3.79: Concept Check For each value of s, use a calculator to find sin s a...
 3.3.80: Concept Check For each value of s, use a calculator to find sin s a...
 3.3.81: Concept Check In Exercises 81 and 82, each graphing calculator scre...
 3.3.82: Concept Check In Exercises 81 and 82, each graphing calculator scre...
 3.3.83: Elevation of the Sun Refer to Example 5. (a) Repeat the example for...
 3.3.84: Length of a Day The number of daylight hours H at any location can ...
 3.3.85: Maximum Temperatures Because the values of the circular functions r...
 3.3.86: Temperature in Fairbanks Suppose the temperature in Fairbanks is mo...
 3.3.87: Refer to Figures 18 and 19. Suppose that angle u measures 60_ . Fin...
 3.3.88: Refer to Figures 18 and 19. Repeat Exercise 87 for u = 38_ , but gi...
Solutions for Chapter 3.3: The Unit Circle and Circular Functions
Full solutions for Trigonometry  10th Edition
ISBN: 9780321671776
Solutions for Chapter 3.3: The Unit Circle and Circular Functions
Get Full SolutionsChapter 3.3: The Unit Circle and Circular Functions includes 88 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Trigonometry, edition: 10. Since 88 problems in chapter 3.3: The Unit Circle and Circular Functions have been answered, more than 35645 students have viewed full stepbystep solutions from this chapter. Trigonometry was written by and is associated to the ISBN: 9780321671776.

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

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.)

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.

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.

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.

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

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

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

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 .

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

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 .

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column 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.

Solvable system Ax = b.
The right side b is in the column space of A.

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

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·