- 7.3.Problem 1: In triangle ABC, A 35, a 4.0, and b 12. Find angle B. C
- 7.3.a: What are the three possibilities for the resulting triangle or tria...
- 7.3.1: The ambiguous case refers to an oblique triangle in which we are gi...
- 7.3.Problem 2: Find the missing parts of triangle ABC if A 42, a 52 cm, and b 64 c...
- 7.3.b: In triangle ABC, A 40, B 48, and B 132. What are the measures of an...
- 7.3.2: With the ambiguous case, there may be _____, _____, or _____ triang...
- 7.3.3: When solving the ambiguous case using the law of sines, the first s...
- 7.3.Problem 3: Find the
- 7.3.c: How many triangles are possible if angle A is greater than 90 and s...
- 7.3.4: When solving the ambiguous case using the law of cosines, the first...
- 7.3.Problem 4: In triangle ABC, if A 36, a 5.0, and b 14, find angle B. E
- 7.3.d: How many triangles are possible if angle A is greater than 90 and s...
- 7.3.5: A 150, b 30 ft, a 10 ft; no solution 6
- 7.3.Problem 5: Find the missing
- 7.3.6: A 30, b 40 ft, a 10 ft; no solution 7
- 7.3.Problem 6: Find the missing parts of triangle ABC if C 32.7, a 115 ft, and c 2...
- 7.3.7: A 120, b 20 cm, a 30 cm; one solution 8
- 7.3.Problem 7: A plane is flying with an airspeed of 165 miles per hour and a head...
- 7.3.8: A 30, b 12 cm, a 6 cm; one solution 9.
- 7.3.9: A 60, b 18 m, a 16 m; two solutions 1
- 7.3.10: A 20, b 40 m, a 30 m; two solutions F
- 7.3.11: A 38, a 41 ft, b 54 ft
- 7.3.12: A 38, a 41 ft, b 54 ft
- 7.3.13: A 112.2, a 43.8 cm, b 22.3 cm 1
- 7.3.14: A 124.3, a 27.3 cm, b 50.2 cm 1
- 7.3.15: C 27 50, c 347 m, b 425 m 1
- 7.3.16: C 51 30, c 707 m, b 821 m 1
- 7.3.17: B 62 40, b 6.78 inches, c 3.48 inches 1
- 7.3.18: B 45 10, b 1.79 inches, c 1.12 inches 1
- 7.3.19: B 118, b 0.68 cm, a 0.92 cm 2
- 7.3.20: B 30, b 4.2 cm, a 8.4 cm 2
- 7.3.21: A 142, b 2.9 yd, a 1.4 yd 2
- 7.3.22: A 65, b 7.6 yd, a 7.1 yd 2
- 7.3.23: C 26.8, c 36.8 km, b 36.8 km 2
- 7.3.24: C 73.4, c 51.1 km, b 92.4 km 2
- 7.3.25: Distance A 51-foot wire running from the top of a tent pole to the ...
- 7.3.26: Distance A hot-air balloon is held at a constant altitude by two ro...
- 7.3.27: Current A ship is headed due north at a constant 16 miles per hour....
- 7.3.28: Ground Speed A plane is headed due east with an airspeed of 340 mil...
- 7.3.29: Ground Speed A ship headed due east is moving through the water at ...
- 7.3.30: True Course A plane headed due east is traveling with an airspeed o...
- 7.3.31: Leaning Windmill After a wind storm, a farmer notices that his 32-f...
- 7.3.32: Distance A boy is riding his motorcycle on a road that runs east an...
- 7.3.33: A sailboat set a course of N 25 E from a small port along a shoreli...
- 7.3.34: In 33, find the shortest (perpendicular) distance to shore from eac...
- 7.3.35: 4 sin csc 0 36
- 7.3.36: 2 sin 1 csc 3
- 7.3.37: 2 cos sin 20 38
- 7.3.38: cos 23 cos 2 0 3
- 7.3.39: 18 sec2 17 tan sec 12 0 40
- 7.3.40: 7 sin2 9 cos 20 F
- 7.3.41: 2 cos x sec x tan x 0
- 7.3.42: 2 cos2 x sin x 1
- 7.3.43: sin x cos x 0
- 7.3.44: sin x cos x 1
- 7.3.45: Use the law of sines to find C for triangle ABC if B 35, a 28 feet,...
- 7.3.46: Given triangle ABC with B 35, a 28 feet, and b 19 feet, if the law ...
- 7.3.47: A plane headed due west is traveling with a constant speed of 214 m...
Solutions for Chapter 7.3: The Ambiguous Case
Full solutions for Trigonometry | 7th Edition
Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.
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.
z = a - ib for any complex number z = a + ib. Then zz = Iz12.
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.
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.
Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA-1 yll2 = Y T(AAT)-1 Y = 1 displayed by eigshow; axis lengths ad
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.
Invert A by row operations on [A I] to reach [I A-I].
A symmetric matrix with eigenvalues of both signs (+ and - ).
lA-II = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n - 1, volume of box = I det( A) I.
Length II x II.
Square root of x T x (Pythagoras in n dimensions).
Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.
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.
Nullspace N (A)
= All solutions to Ax = O. Dimension n - r = (# columns) - rank.
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.
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.
Pseudoinverse A+ (Moore-Penrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).
Similar matrices A and B.
Every B = M-I AM has the same eigenvalues as A.
Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.