 3.3.Problem 1: Use Figure 5 to find the sine, cosine, and tangent of 2/3.
 3.3.a: If (x, y) is any point on the unit circle, and t is the distance fr...
 3.3.1: If (x, y) is a point on the unit circle and t is the distance from ...
 3.3.Problem 2: Use the u
 3.3.b: What is the first step in finding all values of t between 0 and 2on...
 3.3.2: On the unit circle, the _______ measure of a central angle and the ...
 3.3.Problem 3: Find tan t i
 3.3.c: When we evaluate sin 2, how do we know whether to set the calculato...
 3.3.3: The input to a trigonometric function is formally called the ______...
 3.3.Problem 4: Evaluate cos (7/3). Identify the function, the argument, and the va...
 3.3.d: Why are the sine and cosine functions defined for all real numbers?
 3.3.4: The circular functions are functions of _______ numbers, so we must...
 3.3.Problem 5: Evaluate
 3.3.5: The two trigonometric functions having a domain of all real numbers...
 3.3.Problem 6: Determine which statements are possible for some real number z:
 3.3.6: The largest possible value for the sine or cosine function is _____...
 3.3.Problem 7: Describe how tan t varies as t increases from 0 to /2.
 3.3.7: sin 30
 3.3.8: cos 225
 3.3.9: cot 90
 3.3.10: tan 300
 3.3.11: sec 120
 3.3.12: csc 210
 3.3.13: cos
 3.3.14: sin
 3.3.15: tan
 3.3.16: sec 1
 3.3.17: csc
 3.3.18: cot
 3.3.19: 150
 3.3.20: 90
 3.3.21: 5/3
 3.3.22: 11/6
 3.3.23: 180
 3.3.24: 270
 3.3.25: 3/4 2
 3.3.26: 5/4
 3.3.27: sin 1/2 2
 3.3.28: sin 1/2 2
 3.3.29: cos 3/2 30
 3.3.30: cos 0
 3.3.31: tan 3 32
 3.3.32: cot 3
 3.3.33: 4 3
 3.3.34: 4
 3.3.35: 7 6
 3.3.36: 5 12
 3.3.37: cos t
 3.3.38: sin t 3
 3.3.39: sin t 1 4
 3.3.40: cos t 1 1
 3.3.41: 120
 3.3.42: 75
 3.3.43: 225
 3.3.44: 310
 3.3.45: sin t 4
 3.3.46: cos
 3.3.47: sin t co
 3.3.48: sin t co
 3.3.49: If angle is in standard position and the terminal side of intersect...
 3.3.50: If angle is in standard position and the terminal side of intersect...
 3.3.51: If t is the distance from (1, 0) to (0.5403, 0.8415) along the circ...
 3.3.52: If t is the distance from (1, 0) to (0.9422, 0.3350) along the circ...
 3.3.53: Find the coordinates of the point on the unit circle measured 4 uni...
 3.3.54: Find the coordinates of the point on the unit circle measured 5 uni...
 3.3.55: Estimate sin 2.7.
 3.3.56: Estimate cos 2.7.
 3.3.57: Estimate sec 4.6.
 3.3.58: Estimate sec 4.6.
 3.3.59: Estimate tan 3.9.
 3.3.60: Estimate cot 2.2.
 3.3.61: Estimate if cos 0.8 and 0 2. 62.
 3.3.62: Estimate if sin 0.7 and 0 2. Iden
 3.3.63: tan 5
 3.3.64: cos 6
 3.3.65: sin 2
 3.3.66: cos (A B)
 3.3.67: sin x 68
 3.3.68: tan
 3.3.69: Evaluate cos . Identify the function, the argument of the function,...
 3.3.70: Evaluate sin . Identify the function, the argument of the function,...
 3.3.71: sin 4
 3.3.72: cos (1.5) 7
 3.3.73: tan (45)
 3.3.74: cot 30
 3.3.75: sec 0.8
 3.3.76: csc
 3.3.77: sin 278
 3.3.78: cos 2
 3.3.79: sin 28
 3.3.80: cos 2
 3.3.81: Which of the six trigonometric functions are not defined at 0?
 3.3.82: Which of the six trigonometric functions are not defined at /2?
 3.3.83: sec z
 3.3.84: csc 0 z
 3.3.85: cos z 8
 3.3.86: sin 0 z
 3.3.87: tan z
 3.3.88: cot z 8
 3.3.89: sin z 1.2
 3.3.90: cos z
 3.3.91: tan z 9
 3.3.92: cot z 0
 3.3.93: sec z 9
 3.3.94: csc z 0
 3.3.95: Describe how csc t varies as t increases from 0 to /2.
 3.3.96: Describe how cot t varies as t increases from 0 to /2.
 3.3.97: Describe how sin t varies as t increases from /2 to .
 3.3.98: Describe how cos t varies as t increases from /2 to .
 3.3.99: Show why AE tan .
 3.3.100: Show why OF csc . 1
 3.3.101: If is close to 0, determine whether the value of each trigonometric...
 3.3.102: If is close to /2, determine whether the value of each trigonometri...
 3.3.103: Why is sec  1?
 3.3.104: Why is cos  1?
 3.3.105: Determine whether the value of each trigonometric function is great...
 3.3.106: Determine whether the value of each trigonometric function is great...
 3.3.107: A 42, c 36 1
 3.3.108: A 58, c 17
 3.3.109: B 22, b 320 1
 3.3.110: a 1
 3.3.111:
 3.3.112:
 3.3.113: Use the unit circle in Figure 13 to approximate cos 2. a. 0.9 b. 0....
 3.3.114: If angle is in standard position and the terminal side of intersect...
 3.3.115: Use a calculator to approximate sec 4. a. 1.3213 b. 14.3356 c. 1.52...
 3.3.116: Which statement is possible for some real number z? a. sin z 2 b. s...
Solutions for Chapter 3.3: Definition III: Circular Functions
Full solutions for Trigonometry  7th Edition
ISBN: 9781111826857
Solutions for Chapter 3.3: Definition III: Circular Functions
Get Full SolutionsChapter 3.3: Definition III: Circular Functions includes 127 full stepbystep solutions. Trigonometry was written by and is associated to the ISBN: 9781111826857. This textbook survival guide was created for the textbook: Trigonometry, edition: 7. This expansive textbook survival guide covers the following chapters and their solutions. Since 127 problems in chapter 3.3: Definition III: Circular Functions have been answered, more than 24627 students have viewed full stepbystep solutions from this chapter.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

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).

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

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

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

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

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.

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.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

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.

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.

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.

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.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

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

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