 2.1.1: Find exact values or expressions for sin A, cos A, and tan A. See E...
 2.1.2: Find exact values or expressions for sin A, cos A, and tan A. See E...
 2.1.3: Find exact values or expressions for sin A, cos A, and tan A. See E...
 2.1.4: Find exact values or expressions for sin A, cos A, and tan A. See E...
 2.1.5: Concept Check For each trigonometric function in Column I, choose i...
 2.1.6: Concept Check For each trigonometric function in Column I, choose i...
 2.1.7: Concept Check For each trigonometric function in Column I, choose i...
 2.1.8: Concept Check For each trigonometric function in Column I, choose i...
 2.1.9: Concept Check For each trigonometric function in Column I, choose i...
 2.1.10: Concept Check For each trigonometric function in Column I, choose i...
 2.1.11: Suppose ABC is a right triangle with sides of lengths a, b, and c a...
 2.1.12: Suppose ABC is a right triangle with sides of lengths a, b, and c a...
 2.1.13: Suppose ABC is a right triangle with sides of lengths a, b, and c a...
 2.1.14: Suppose ABC is a right triangle with sides of lengths a, b, and c a...
 2.1.15: Suppose ABC is a right triangle with sides of lengths a, b, and c a...
 2.1.16: Suppose ABC is a right triangle with sides of lengths a, b, and c a...
 2.1.17: Suppose ABC is a right triangle with sides of lengths a, b, and c a...
 2.1.18: Suppose ABC is a right triangle with sides of lengths a, b, and c a...
 2.1.19: Suppose ABC is a right triangle with sides of lengths a, b, and c a...
 2.1.20: Concept Check Give a summary of the six cofunction relationships.
 2.1.21: Write each function in terms of its cofunction. Assume that all ang...
 2.1.22: Write each function in terms of its cofunction. Assume that all ang...
 2.1.23: Write each function in terms of its cofunction. Assume that all ang...
 2.1.24: Write each function in terms of its cofunction. Assume that all ang...
 2.1.25: Write each function in terms of its cofunction. Assume that all ang...
 2.1.26: Write each function in terms of its cofunction. Assume that all ang...
 2.1.27: Write each function in terms of its cofunction. Assume that all ang...
 2.1.28: Write each function in terms of its cofunction. Assume that all ang...
 2.1.29: Write each function in terms of its cofunction. Assume that all ang...
 2.1.30: With a calculator, evaluate sin190  u2 and cos u for various value...
 2.1.31: Find one solution for each equation. Assume that all angles in whic...
 2.1.32: Find one solution for each equation. Assume that all angles in whic...
 2.1.33: Find one solution for each equation. Assume that all angles in whic...
 2.1.34: Find one solution for each equation. Assume that all angles in whic...
 2.1.35: Find one solution for each equation. Assume that all angles in whic...
 2.1.36: Find one solution for each equation. Assume that all angles in whic...
 2.1.37: Find one solution for each equation. Assume that all angles in whic...
 2.1.38: Find one solution for each equation. Assume that all angles in whic...
 2.1.39: Find one solution for each equation. Assume that all angles in whic...
 2.1.40: Find one solution for each equation. Assume that all angles in whic...
 2.1.41: Determine whether each statement is true or false. See Example 4. s...
 2.1.42: Determine whether each statement is true or false. See Example 4. t...
 2.1.43: Determine whether each statement is true or false. See Example 4. s...
 2.1.44: Determine whether each statement is true or false. See Example 4. c...
 2.1.45: Determine whether each statement is true or false. See Example 4. t...
 2.1.46: Determine whether each statement is true or false. See Example 4. c...
 2.1.47: Determine whether each statement is true or false. See Example 4. s...
 2.1.48: Determine whether each statement is true or false. See Example 4. c...
 2.1.49: For each expression, give the exact value. See Example 5. tan 30
 2.1.50: For each expression, give the exact value. See Example 5. cot 30
 2.1.51: For each expression, give the exact value. See Example 5. sin 30
 2.1.52: For each expression, give the exact value. See Example 5. cos 30
 2.1.53: For each expression, give the exact value. See Example 5. sec 30
 2.1.54: For each expression, give the exact value. See Example 5. csc 30
 2.1.55: For each expression, give the exact value. See Example 5. csc 45
 2.1.56: For each expression, give the exact value. See Example 5. sec 45
 2.1.57: For each expression, give the exact value. See Example 5. cos 45
 2.1.58: For each expression, give the exact value. See Example 5. cot 45
 2.1.59: For each expression, give the exact value. See Example 5. tan 45
 2.1.60: For each expression, give the exact value. See Example 5. sin 45
 2.1.61: For each expression, give the exact value. See Example 5. sin 60
 2.1.62: For each expression, give the exact value. See Example 5. cos 60
 2.1.63: For each expression, give the exact value. See Example 5. tan 60
 2.1.64: For each expression, give the exact value. See Example 5. csc 60
 2.1.65: Sketch a line segment from P perpendicular to the xaxis.
 2.1.66: Use the trigonometric ratios for a 45 angle to label the sides of t...
 2.1.67: Which sides of the right triangle give the coordinates of point P? ...
 2.1.68: The figure at the right shows a 60 central angle in a circle of rad...
 2.1.69: Concept Check Refer to the table. What trigonometric functions are ...
 2.1.70: Concept Check Refer to the table. What trigonometric functions are ...
 2.1.71: Concept Check What value of A between 0 and 90 will produce the out...
 2.1.72: A student was asked to give the exact value of sin 45. Using a calc...
 2.1.73: With a graphing calculator, find the coordinates of the point of in...
 2.1.74: Find the equation of the line that passes through the origin and ma...
 2.1.75: Find the equation of the line that passes through the origin and ma...
 2.1.76: What angle does the line y = 23 3 x make with the positive xaxis?
 2.1.77: What angle does the line y = 23x make with the positive xaxis?
 2.1.78: Construct a square with each side of length k. (a) Draw a diagonal ...
 2.1.79: Construct an equilateral triangle with each side having length 2k. ...
 2.1.80: Find the exact value of each part labeled with a variable in each f...
 2.1.81: Find the exact value of each part labeled with a variable in each f...
 2.1.82: Find the exact value of each part labeled with a variable in each f...
 2.1.83: Find the exact value of each part labeled with a variable in each f...
 2.1.84: Find a formula for the area of each figure in terms of s.
 2.1.85: Find a formula for the area of each figure in terms of s.
 2.1.86: Concept Check Suppose you know the length of one side and one acute...
Solutions for Chapter 2.1: Trigonometric Functions of Acute Angles
Full solutions for Trigonometry  10th Edition
ISBN: 9780321671776
Solutions for Chapter 2.1: Trigonometric Functions of Acute Angles
Get Full SolutionsChapter 2.1: Trigonometric Functions of Acute Angles includes 86 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Trigonometry was written by and is associated to the ISBN: 9780321671776. This textbook survival guide was created for the textbook: Trigonometry, edition: 10. Since 86 problems in chapter 2.1: Trigonometric Functions of Acute Angles have been answered, more than 34370 students have viewed full stepbystep solutions from this chapter.

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

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.

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

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.

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

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

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.

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

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

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.

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

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

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.

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

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

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).