 1.3.1.1.157: Find all six trigonometric functions of (J ifthe given point is on ...
 1.3.1.1.158: Find all six trigonometric functions of (J ifthe given point is on ...
 1.3.1.1.159: Find all six trigonometric functions of (J ifthe given point is on ...
 1.3.1.1.160: Find all six trigonometric functions of (J ifthe given point is on ...
 1.3.1.1.161: Find all six trigonometric functions of (J ifthe given point is on ...
 1.3.1.1.162: Find all six trigonometric functions of (J ifthe given point is on ...
 1.3.1.1.163: Find all six trigonometric functions of (J ifthe given point is on ...
 1.3.1.1.164: Find all six trigonometric functions of (J ifthe given point is on ...
 1.3.1.1.165: Find all six trigonometric functions of (J ifthe given point is on ...
 1.3.1.1.166: Find all six trigonometric functions of (J ifthe given point is on ...
 1.3.1.1.167: Find all six trigonometric functions of (J ifthe given point is on ...
 1.3.1.1.168: Find all six trigonometric functions of (J ifthe given point is on ...
 1.3.1.1.169: In the diagrams below, angle (J is in standard position. In each ca...
 1.3.1.1.170: In the diagrams below, angle (J is in standard position. In each ca...
 1.3.1.1.171: In the diagrams below, angle (J is in standard position. In each ca...
 1.3.1.1.172: In the diagrams below, angle (J is in standard position. In each ca...
 1.3.1.1.173: Use your calculator to find sin (J and cos (J if the point (9.36, 7...
 1.3.1.1.174: Use your calculator to find sin 8 and cos 8 if the point (6.36, 2.6...
 1.3.1.1.175: Draw each of the following angles in standard position, find a poin...
 1.3.1.1.176: Draw each of the following angles in standard position, find a poin...
 1.3.1.1.177: Draw each of the following angles in standard position, find a poin...
 1.3.1.1.178: Draw each of the following angles in standard position, find a poin...
 1.3.1.1.179: Draw each of the following angles in standard position, find a poin...
 1.3.1.1.180: Draw each of the following angles in standard position, find a poin...
 1.3.1.1.181: Draw each of the following angles in standard position, find a poin...
 1.3.1.1.182: Draw each of the following angles in standard position, find a poin...
 1.3.1.1.183: Determine whether each statement is true or false. cos 35 < cos 45
 1.3.1.1.184: Determine whether each statement is true or false. sin 35 < sin 45
 1.3.1.1.185: Determine whether each statement is true or false. sec 60 < sec 75
 1.3.1.1.186: Determine whether each statement is true or false. cot 60 < cot 75
 1.3.1.1.187: Use Definition I and Figure 1 to answer the following. Explain why ...
 1.3.1.1.188: Use Definition I and Figure 1 to answer the following. Explain why ...
 1.3.1.1.189: Use Definition I and Figure 1 to answer the following. Why is lese ...
 1.3.1.1.190: Use Definition I and Figure 1 to answer the following. Why is cos (...
 1.3.1.1.191: Use Definition I and Figure 1 to answer the following. As (J increa...
 1.3.1.1.192: Use Definition I and Figure 1 to answer the following. As (J increa...
 1.3.1.1.193: Use Definition I and Figure 1 to answer the following. As () increa...
 1.3.1.1.194: Use Definition I and Figure 1 to answer the following. As (J increa...
 1.3.1.1.195: Indicate the two quadrants () could terminate in if sin () = ~
 1.3.1.1.196: Indicate the two quadrants () could terminate in if cos (J = t
 1.3.1.1.197: Indicate the two quadrants () could terminate in if cos () = 0.45
 1.3.1.1.198: Indicate the two quadrants () could terminate in if sin (J 3A1iO
 1.3.1.1.199: Indicate the two quadrants () could terminate in if tan (J
 1.3.1.1.200: Indicate the two quadrants () could terminate in if cot (J 20
 1.3.1.1.201: Indicate the two quadrants () could terminate in if esc (J  2.45
 1.3.1.1.202: Indicate the two quadrants () could terminate in if ec (J = 2
 1.3.1.1.203: Indicate the quadrants in which the terminal side of () must lie in...
 1.3.1.1.204: Indicate the quadrants in which the terminal side of () must lie in...
 1.3.1.1.205: Indicate the quadrants in which the terminal side of () must lie in...
 1.3.1.1.206: Indicate the quadrants in which the terminal side of () must lie in...
 1.3.1.1.207: For 51 through 64, find the remaining trigonometric functions of (J...
 1.3.1.1.208: For 51 through 64, find the remaining trigonometric functions of (J...
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 1.3.1.1.219: For 51 through 64, find the remaining trigonometric functions of (J...
 1.3.1.1.220: For 51 through 64, find the remaining trigonometric functions of (J...
 1.3.1.1.221: For 51 through 64, find the remaining trigonometric functions of (J...
 1.3.1.1.222: For 51 through 64, find the remaining trigonometric functions of (J...
 1.3.1.1.223: For 51 through 64, find the remaining trigonometric functions of (J...
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 1.3.1.1.225: For 51 through 64, find the remaining trigonometric functions of (J...
 1.3.1.1.226: For 51 through 64, find the remaining trigonometric functions of (J...
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 1.3.1.1.232: For 51 through 64, find the remaining trigonometric functions of (J...
Solutions for Chapter 1.3: THE SIX TRIGONOMETRIC FUNCTIONS
Full solutions for Trigonometry
ISBN: 9780495108351
Solutions for Chapter 1.3: THE SIX TRIGONOMETRIC FUNCTIONS
Get Full SolutionsChapter 1.3: THE SIX TRIGONOMETRIC FUNCTIONS includes 76 full stepbystep solutions. Since 76 problems in chapter 1.3: THE SIX TRIGONOMETRIC FUNCTIONS have been answered, more than 29901 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Trigonometry, edition: . This expansive textbook survival guide covers the following chapters and their solutions. Trigonometry was written by and is associated to the ISBN: 9780495108351.

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.)

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.

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

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 .

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

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

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

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

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.