 7.1: If A 32, B 70, and a 3.8 inches, use the law of sines to find b. 2
 7.2: If A 38.2, B 63.4, and c 42.0 cm, find all the missing parts.
 7.3: If A 24.7, C 106.1, and b 34.0 cm, find all the missing parts. 4
 7.4: If C 60, a 10 cm, and b 12 cm, use the law of cosines to find c.
 7.5: If a 5 km, b 7 km, and c 9 km, use the law of cosines to find C to ...
 7.6: Find all the missing parts if a 6.4 m, b 2.8 m, and C 119.
 7.7: Find all the missing parts if b 3.7 m, c 6.2 m, and A 35.
 7.8: Use the law of sines to show that no triangle exists for which A 60...
 7.9: Use the law of sines to show that exactly one triangle exists for w...
 7.10: Find two triangles for which A 51, a 6.5 ft, and b 7.9 ft. 1
 7.11: Find the area of the triangle in 2.
 7.12: Find the area of the triangle in 4.
 7.13: Find the area of the triangle in 5.
 7.14: Geometry The two equal sides of an isosceles triangle are each 38 c...
 7.15: Angle of Elevation A man standing near a building notices that the ...
 7.16: Geometry The diagonals of a parallelogram are 26.8 meters and 39.4 ...
 7.17: Arc Length Suppose Figure 1 is an exaggerated diagram of a plane fl...
 7.18: Distance and Bearing A man wandering in the desert walks 3.3 miles ...
 7.19: Distance Two guy wires from the top of a tent pole are anchored to ...
 7.20: Ground Speed A plane is headed due east with an airspeed of 345 mil...
 7.21: Height of a Tree To estimate the height of a tree, two people posit...
 7.22: True Course and Speed A plane flying with an airspeed of 325 miles ...
 7.23: U
 7.24: 3U 5V
 7.25: 2V W
 7.26: V W
 7.27: The angle between U and V to the nearest tenth of a degree.
 7.28: Show that V 3i 6j and W 8i 4j are perpendicular.
 7.29: Find the value of b so that vectors U 5i 12j and V 4i bj are perpen...
 7.30: Find the work performed by the force F 33i 4j in moving an object, ...
Solutions for Chapter 7: Triangles
Full solutions for Trigonometry  7th Edition
ISBN: 9781111826857
Solutions for Chapter 7: Triangles
Get Full SolutionsChapter 7: Triangles includes 30 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 30 problems in chapter 7: Triangles have been answered, more than 28137 students have viewed full stepbystep solutions from this chapter.

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.

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

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.

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.

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

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.

Outer product uv T
= column times row = rank one matrix.

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.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

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

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

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·

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.