 7.3.1: Concept Check Assume a triangle ABC has standard labeling. (a) Dete...
 7.3.2: Concept Check Assume a triangle ABC has standard labeling. (a) Dete...
 7.3.3: Concept Check Assume a triangle ABC has standard labeling. (a) Dete...
 7.3.4: Concept Check Assume a triangle ABC has standard labeling. (a) Dete...
 7.3.5: Concept Check Assume a triangle ABC has standard labeling. (a) Dete...
 7.3.6: Concept Check Assume a triangle ABC has standard labeling. (a) Dete...
 7.3.7: Concept Check Assume a triangle ABC has standard labeling. (a) Dete...
 7.3.8: Concept Check Assume a triangle ABC has standard labeling. (a) Dete...
 7.3.9: Find the length of the remaining side of each triangle. Do not use ...
 7.3.10: Find the length of the remaining side of each triangle. Do not use ...
 7.3.11: Find the measure of u in each triangle. Do not use a calculator.
 7.3.12: Find the measure of u in each triangle. Do not use a calculator.
 7.3.13: Solve each triangle. Approximate values to the nearest tenth.
 7.3.14: Solve each triangle. Approximate values to the nearest tenth.
 7.3.15: Solve each triangle. Approximate values to the nearest tenth.
 7.3.16: Solve each triangle. Approximate values to the nearest tenth.
 7.3.17: Solve each triangle. Approximate values to the nearest tenth.
 7.3.18: Solve each triangle. Approximate values to the nearest tenth.
 7.3.19: Solve each triangle. See Examples 2 and 3. A = 41.4_, b = 2.78 yd, ...
 7.3.20: Solve each triangle. See Examples 2 and 3. C = 28.3_, b = 5.71 in.,...
 7.3.21: Solve each triangle. See Examples 2 and 3. C = 45.6_, b = 8.94 m, a...
 7.3.22: Solve each triangle. See Examples 2 and 3. A = 67.3_, b = 37.9 km, ...
 7.3.23: Solve each triangle. See Examples 2 and 3. a = 9.3 cm, b = 5.7 cm, ...
 7.3.24: Solve each triangle. See Examples 2 and 3. a = 28 ft, b = 47 ft, c ...
 7.3.25: Solve each triangle. See Examples 2 and 3. a = 42.9 m, b = 37.6 m, ...
 7.3.26: Solve each triangle. See Examples 2 and 3. a = 189 yd, b = 214 yd, ...
 7.3.27: Solve each triangle. See Examples 2 and 3. a = 965 ft, b = 876 ft, ...
 7.3.28: Solve each triangle. See Examples 2 and 3. a = 324 m, b = 421 m, c ...
 7.3.29: Solve each triangle. See Examples 2 and 3. A = 80_ 40_, b = 143 cm,...
 7.3.30: Solve each triangle. See Examples 2 and 3. C = 72_ 40_, a = 327 ft,...
 7.3.31: Solve each triangle. See Examples 2 and 3. B = 74.8_, a = 8.92 in.,...
 7.3.32: Solve each triangle. See Examples 2 and 3. C = 59.7_, a = 3.73 mi, ...
 7.3.33: Solve each triangle. See Examples 2 and 3. A = 112.8_, b = 6.28 m, ...
 7.3.34: Solve each triangle. See Examples 2 and 3. B = 168.2_, a = 15.1 cm,...
 7.3.35: Solve each triangle. See Examples 2 and 3. a = 3.0 ft, b = 5.0 ft, ...
 7.3.36: Solve each triangle. See Examples 2 and 3. a = 4.0 ft, b = 5.0 ft, ...
 7.3.37: Refer to Figure 12. If you attempt to find any angle of a triangle ...
 7.3.38: The shortest distance between two points is a straight line. Explai...
 7.3.39: Distance across a River Points A and B are on opposite sides of Fal...
 7.3.40: Distance across a Ravine Points X and Y are on opposite sides of a ...
 7.3.41: Angle in a Parallelogram A parallelogram has sides of length 25.9 c...
 7.3.42: Diagonals of a Parallelogram The sides of a parallelogram are 4.0 c...
 7.3.43: Flight Distance Airports A and B are 450 km apart, on an eastwest ...
 7.3.44: Distance Traveled by a Plane An airplane flies 180 mi from point X ...
 7.3.45: Distance between Ends of the Vietnam Memorial The Vietnam Veterans ...
 7.3.46: Distance between Two Ships Two ships leave a harbor together, trave...
 7.3.47: Distance between a Ship and a Rock A ship is sailing east. At one p...
 7.3.48: Distance between a Ship and a Submarine From an airplane flying ove...
 7.3.49: Truss Construction A triangular truss is shown in the figure. Find ...
 7.3.50: Truss Construction Find angle b in the truss shown in the figure. 5
 7.3.51: Distance between a Beam and Cables A weight is supported by cables ...
 7.3.52: Distance between Points on a Crane A crane with a counterweight is ...
 7.3.53: Distance on a Baseball Diamond A baseball diamond is a square, 90.0...
 7.3.54: Distance on a Softball Diamond A softball diamond is a square, 60.0...
 7.3.55: Distance between a Ship and a Point Starting at point A, a ship sai...
 7.3.56: Distance between Two Factories Two factories blow their whistles at...
 7.3.57: Measurement Using Triangulation Surveyors are often confronted with...
 7.3.58: Path of a Ship A ship sailing due east in the North Atlantic has be...
 7.3.59: Length of a Tunnel To measure the distance through a mountain for a...
 7.3.60: Distance between an Airplane and a Mountain A person in a plane fly...
 7.3.61: Find the measure of each angle u to two decimal places.
 7.3.62: Find the measure of each angle u to two decimal places.
 7.3.63: Find the exact area of each triangle using the formula = 12 bh, and...
 7.3.64: Find the exact area of each triangle using the formula = 12 bh, and...
 7.3.65: Find the area of each triangle ABC. See Example 5. a = 12 m, b = 16...
 7.3.66: Find the area of each triangle ABC. See Example 5. a = 22 in., b = ...
 7.3.67: Find the area of each triangle ABC. See Example 5. a = 154 cm, b = ...
 7.3.68: Find the area of each triangle ABC. See Example 5. a = 25.4 yd, b =...
 7.3.69: Find the area of each triangle ABC. See Example 5. a = 76.3 ft, b =...
 7.3.70: Find the area of each triangle ABC. See Example 5. a = 15.89 m, b =...
 7.3.71: Perfect Triangles A perfect triangle is a triangle whose sides have...
 7.3.72: Heron Triangles A Heron triangle is a triangle having integer sides...
 7.3.73: Area of the Bermuda Triangle Find the area of the Bermuda Triangle ...
 7.3.74: Required Amount of Paint A painter needs to cover a triangular regi...
 7.3.75: Consider triangle ABC shown here. (a) Use the law of sines to find ...
 7.3.76: Show that the measure of angle A is twice the measure of angle B. (...
 7.3.77: Draw a triangle with vertices A12, 52, B11, 32, and C14, 02, and u...
 7.3.78: Find the area of triangle ABC using formula (b). (First use the law...
 7.3.79: Find the area of triangle ABC using formula (c)that is, Herons form...
 7.3.80: Find the area of triangle ABC using new formula (d).
Solutions for Chapter 7.3: The Law of Cosines
Full solutions for Trigonometry  10th Edition
ISBN: 9780321671776
Solutions for Chapter 7.3: The Law of Cosines
Get Full SolutionsTrigonometry was written by and is associated to the ISBN: 9780321671776. Chapter 7.3: The Law of Cosines includes 80 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 80 problems in chapter 7.3: The Law of Cosines have been answered, more than 35645 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Trigonometry, edition: 10.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

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.

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

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.

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

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.

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

lAII = 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.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

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.

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

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·

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.