 2.1: Find exact values of the six trigonometric functions for each angle A.
 2.2: Find exact values of the six trigonometric functions for each angle A.
 2.3: Find one solution for each equation. Assume that all angles involve...
 2.4: Find one solution for each equation. Assume that all angles involve...
 2.5: Find one solution for each equation. Assume that all angles involve...
 2.6: Find one solution for each equation. Assume that all angles involve...
 2.7: Determine whether each statement is true or false. If false, tell w...
 2.8: Determine whether each statement is true or false. If false, tell w...
 2.9: Determine whether each statement is true or false. If false, tell w...
 2.10: Determine whether each statement is true or false. If false, tell w...
 2.11: Explain why, in the figure, the cosine of angle A is equal to the s...
 2.12: Which one of the following cannot be exactly determined using the m...
 2.13: Find exact values of the six trigonometric functions for each angle...
 2.14: Find exact values of the six trigonometric functions for each angle...
 2.15: Find exact values of the six trigonometric functions for each angle...
 2.16: Find exact values of the six trigonometric functions for each angle...
 2.17: Find all values of u, if u is in the interval 30, 3602 and u has th...
 2.18: Find all values of u, if u is in the interval 30, 3602 and u has th...
 2.19: Find all values of u, if u is in the interval 30, 3602 and u has th...
 2.20: Find all values of u, if u is in the interval 30, 3602 and u has th...
 2.21: Evaluate each expression. Give exact values. tan2 120  2 cot 240
 2.22: Evaluate each expression. Give exact values. cos 60 + 2 sin2 30
 2.23: Evaluate each expression. Give exact values. sec2 300  2 cos2 150
 2.24: Find the sine, cosine, and tangent function values for each angle.
 2.25: Use a calculator to approximate the value of each expression. Give ...
 2.26: Use a calculator to approximate the value of each expression. Give ...
 2.27: Use a calculator to approximate the value of each expression. Give ...
 2.28: Use a calculator to approximate the value of each expression. Give ...
 2.29: Use a calculator to approximate the value of each expression. Give ...
 2.30: Use a calculator to approximate the value of each expression. Give ...
 2.31: Use a calculator to find each value of u, where u is in the interva...
 2.32: Use a calculator to find each value of u, where u is in the interva...
 2.33: Use a calculator to find each value of u, where u is in the interva...
 2.34: Use a calculator to find each value of u, where u is in the interva...
 2.35: Use a calculator to find each value of u, where u is in the interva...
 2.36: Use a calculator to find each value of u, where u is in the interva...
 2.37: Find two angles in the interval 30, 3602 that satisfy each of the f...
 2.38: Find two angles in the interval 30, 3602 that satisfy each of the f...
 2.39: Determine whether each statement is true or false. If false, tell w...
 2.40: Determine whether each statement is true or false. If false, tell w...
 2.41: Determine whether each statement is true or false. If false, tell w...
 2.42: Determine whether each statement is true or false. If false, tell w...
 2.43: A student wants to use a calculator to find the value of cot 25. Ho...
 2.44: Explain the process for using a calculator to find sec1 10.
 2.45: Solve each right triangle. In Exercise 46, give angles to the neare...
 2.46: Solve each right triangle. In Exercise 46, give angles to the neare...
 2.47: Solve each right triangle. In Exercise 46, give angles to the neare...
 2.48: Solve each right triangle. In Exercise 46, give angles to the neare...
 2.49: Solve each problem. (Source for Exercises 49 and 50: Parker, M., Ed...
 2.50: (Modeling) Double Vision To correct mild double vision, a small amo...
 2.51: Height of a Tower The angle of elevation from a point 93.2 ft from ...
 2.52: Height of a Tower The angle of depression from a television tower t...
 2.53: Length of a Diagonal One side of a rectangle measures 15.24 cm. The...
 2.54: Length of Sides of an Isosceles Triangle An isosceles triangle has ...
 2.55: Distance between Two Points The bearing of point B from point C is ...
 2.56: Distance a Ship Sails The bearing from point A to point B is S 55 E...
 2.57: Distance between Two Points Two cars leave an intersection at the s...
 2.58: Find a formula for h in terms of k, A, and B. Assume A 6 B
 2.59: Create a right triangle problem whose solution is 3 tan 25
 2.60: Create a right triangle problem whose solution can be found by eval...
 2.61: (Modeling) Height of a Satellite Artificial satellites that orbit E...
 2.62: (Modeling) Fundamental Surveying first fundamental problem of surve...
Solutions for Chapter 2: Acute Angles and Right Triangles
Full solutions for Trigonometry  11th Edition
ISBN: 9780134217437
Solutions for Chapter 2: Acute Angles and Right Triangles
Get Full SolutionsChapter 2: Acute Angles and Right Triangles includes 62 full stepbystep solutions. This textbook survival guide was created for the textbook: Trigonometry, edition: 11. This expansive textbook survival guide covers the following chapters and their solutions. Trigonometry was written by and is associated to the ISBN: 9780134217437. Since 62 problems in chapter 2: Acute Angles and Right Triangles have been answered, more than 20331 students have viewed full stepbystep solutions from this chapter.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

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

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Cyclic shift
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.

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.

Fundamental Theorem.
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.

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

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.

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

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).

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.

Pseudoinverse A+ (MoorePenrose 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).

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.