- 2.1.Problem 1: Triangle ABC is a right triangle with C 90. If a 12 and c 15, find ...
- 2.1.a: In a right triangle, which side is opposite the right angle?
- 2.1.1: Based on its Greek origins, the word trigonometry can be translated...
- 2.1.Problem 2: Use Definition II
- 2.1.b: If A is an acute angle in a right triangle, how do you define sin A...
- 2.1.2: Using Definition II and Figure 8, we would refer to a as the side _...
- 2.1.Problem 3: Fill in the blanks so that each expression becomes a true statement...
- 2.1.c: State the Cofunction Theorem.
- 2.1.3: A trigonometric function of an angle is always equal to the cofunct...
- 2.1.Problem 4: Use exact
- 2.1.d: How are sin 30 and cos 60 related?
- 2.1.4: Match each trigonometric function with its cofunction.
- 2.1.Problem 5: Let x = 60 and y = 45 in each of the following expressions, and the...
- 2.1.5: b 3, c 5 6
- 2.1.6: b 5, c 13
- 2.1.7: a 2, b 1
- 2.1.8: a 3, b 2
- 2.1.9: a 2, b 5
- 2.1.10: a 3, b 7 In
- 2.1.11: B C
- 2.1.12: 3 4 A
- 2.1.13: A
- 2.1.14: A B C
- 2.1.15: A B
- 2.1.16: A B
- 2.1.17: A B
- 2.1.18: A C
- 2.1.19: yA CB
- 2.1.20: y A C
- 2.1.21: Use Definition II to explain why, for any acute angle , it is impos...
- 2.1.22: Use Definition II to explain why, for any acute angle , it is impos...
- 2.1.23: Use Definition II to explain why it is possible to find an angle th...
- 2.1.24: Use Definition II to explain why it is possible to find an angle th...
- 2.1.25: sin 10 cos
- 2.1.26: cos 40 sin
- 2.1.27: tan 8 cot
- 2.1.28: cot 12 tan
- 2.1.29: sin x cos
- 2.1.30: sin y cos
- 2.1.31: tan (90 x) cot
- 2.1.32: tan (90 y) cot
- 2.1.33: x sin x csc x
- 2.1.34: x cos x sec x
- 2.1.35: 4 sin 30
- 2.1.36: 5 sin2 30
- 2.1.37: (2 cos 30)2
- 2.1.38: sin3 30
- 2.1.39: sin2 60 cos2 60
- 2.1.40: (sin 60 cos 60)2
- 2.1.41: sin2 45 2 sin 45 cos 45 cos2 45
- 2.1.42: (sin 45 cos 45)2
- 2.1.43: (tan 45 tan 60)2
- 2.1.44: tan2 45 tan2 60
- 2.1.45: 2 sin x
- 2.1.46: 4 cos y
- 2.1.47: 4 cos(z 30)
- 2.1.48: 2 sin(y 45)
- 2.1.49: 3 sin 2x
- 2.1.50: 3 sin 2y
- 2.1.51: 2 cos(3x 45)
- 2.1.52: 2 sin(90 z)
- 2.1.53: sec 30
- 2.1.54: csc 30
- 2.1.55: csc 60
- 2.1.56: sec 60
- 2.1.57: cot 45
- 2.1.58: cot 30
- 2.1.59: sec 45
- 2.1.60: csc 45
- 2.1.61: cot 60
- 2.1.62: sec 0
- 2.1.63: csc 90
- 2.1.64: cot 90
- 2.1.65: b 8.88, c 9.62 6
- 2.1.66: a 3.42, c 5.70
- 2.1.67: a 19.44, b 5.67 6
- 2.1.68: a 11.28, b 8.46
- 2.1.69: Suppose each edge of the cube shown in Figure 9 is 5 inches long. F...
- 2.1.70: Suppose each edge of the cube shown in Figure 9 is 3 inches long. F...
- 2.1.71: Suppose each edge of the cube shown in Figure 9 is x inches long. F...
- 2.1.72: Suppose each edge of the cube shown in Figure 9 is y inches long. F...
- 2.1.73: Find the distance between the points (3, 2) and (1, 4).
- 2.1.74: Find x so that the distance between (x, 2) and (1, 5) is 13.
- 2.1.75: 135
- 2.1.76: 45
- 2.1.77: 135
- 2.1.78: 210
- 2.1.79: Triangle ABC is a right triangle with C 90. If a 16 and c 20, what ...
- 2.1.80: According to the Cofunction Theorem, which value is equal to sin 35...
- 2.1.81: Which of the following statements is false? a. sin 30 b. sin 0 0 c....
- 2.1.82: Use exact values to simplify 4 cos2 30 2 sin 30. a. 1 3 b. 3 c. 4 d...
Solutions for Chapter 2.1: Definition II: Right Triangle Trigonometry
Full solutions for Trigonometry | 7th Edition
Tv = Av + Vo = linear transformation plus shift.
Remove row i and column j; multiply the determinant by (-I)i + j •
Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.
cond(A) = c(A) = IIAIlIIA-III = 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.
Diagonal matrix D.
dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.
Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).
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.
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 Fn-1c can be computed with ne/2 multiplications. Revolutionary.
Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.
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.
Jordan form 1 = M- 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.
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 .
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.
Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.
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.
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.
Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.
Skew-symmetric matrix K.
The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!