 57.1: Consider the angle = 150. (a) Sketch the angle in standard position...
 57.2: Convert the angle = 2.55 radians to degrees. Round your answer to o...
 57.3: Find cos when tan = 21 20 and sin < 0
 57.4: In Exercises 46, sketch the graph of the function by hand. (Include...
 57.5: In Exercises 46, sketch the graph of the function by hand. (Include...
 57.6: In Exercises 46, sketch the graph of the function by hand. (Include...
 57.7: Find values of a, b, and c such that the graph of the function h(x)...
 57.8: In Exercises 8 and 9, find the exact value of the expression, if po...
 57.9: In Exercises 8 and 9, find the exact value of the expression, if po...
 57.10: Write an algebraic expression equivalent to sin(arccos 2x).
 57.11: Subtract and simplify: sin 1 cos cos sin 1 .
 57.12: In Exercises 1214, verify the identity. 12. cot2 (sec2 1) = 1 13. s...
 57.13: In Exercises 1214, verify the identity. 12. cot2 (sec2 1) = 1 13. s...
 57.14: In Exercises 1214, verify the identity. 12. cot2 (sec2 1) = 1 13. s...
 57.15: In Exercises 15 and 16, solve the equation. 15. sin2 x + 2 sin x + ...
 57.16: In Exercises 15 and 16, solve the equation. 15. sin2 x + 2 sin x + ...
 57.17: Approximate the solutions of the equation cos2 x 5 cos x 1 = 0 in t...
 57.18: In Exercises 18 and 19, use a graphing utility to graph the functio...
 57.19: In Exercises 18 and 19, use a graphing utility to graph the functio...
 57.20: Given that sin u = 12 13, cos v = 3 5, and angles u and v are both ...
 57.21: If tan = 1 2 , find the exact value of tan 2, 0 < < 2 .
 57.22: If tan = 4 3 , find the exact value of sin 2 , < < 3 2 .
 57.23: Write cos 8x + cos 4x as a product.
 57.24: In Exercises 24 27, verify the identity. 24. tan x(1 sin2 x) = 1 2 ...
 57.25: In Exercises 24 27, verify the identity. 24. tan x(1 sin2 x) = 1 2 ...
 57.26: In Exercises 24 27, verify the identity. 24. tan x(1 sin2 x) = 1 2 ...
 57.27: In Exercises 24 27, verify the identity. 24. tan x(1 sin2 x) = 1 2 ...
 57.28: In Exercises 2831, use the information to solve the triangle shown ...
 57.29: In Exercises 2831, use the information to solve the triangle shown ...
 57.30: In Exercises 2831, use the information to solve the triangle shown ...
 57.31: In Exercises 2831, use the information to solve the triangle shown ...
 57.32: Two sides of a triangle have lengths 14 inches and 19 inches. Their...
 57.33: Find the area of a triangle with sides of lengths 30 meters, 41 met...
 57.34: Write the vector u = 3, 5 as a linear combination of the standard u...
 57.35: Find a unit vector in the direction of v = 4i 2j.
 57.36: Find u v for u = 3i + 4j and v = 6i 9j.
 57.37: Find k such that u = i + 2kj and v = 2i j are orthogonal.
 57.38: Find the projection of u = 8, 2 onto v = 1, 5. Then write u as the ...
 57.39: Find the trigonometric form of the complex number plotted at the ri...
 57.40: Write the complex number 63(cos 5 6 + i sin 5 6 ) in standard form.
 57.41: Find the product [4(cos 30 + i sin 30)][6(cos 120 + i sin 120)]. Wr...
 57.42: Find the square roots of 2 + i.
 57.43: Find the three cube roots of 1.
 57.44: Write all the solutions of the equation x4 + 1296 = 0
 57.45: From a point 200 feet from a flagpole, the angles of elevation to t...
 57.46: Write a model for a particle in simple harmonic motion with a maxim...
 57.47: An airplanes velocity with respect to the air is 500 kilometers per...
 57.48: Forces of 60 pounds and 100 pounds have a resultant force of 125 po...
Solutions for Chapter 57: Additional Topics in Trigonometry
Full solutions for Algebra and Trigonometry: Real Mathematics, Real People  7th Edition
ISBN: 9781305071735
Solutions for Chapter 57: Additional Topics in Trigonometry
Get Full SolutionsThis textbook survival guide was created for the textbook: Algebra and Trigonometry: Real Mathematics, Real People, edition: 7. Chapter 57: Additional Topics in Trigonometry includes 48 full stepbystep solutions. Algebra and Trigonometry: Real Mathematics, Real People was written by and is associated to the ISBN: 9781305071735. This expansive textbook survival guide covers the following chapters and their solutions. Since 48 problems in chapter 57: Additional Topics in Trigonometry have been answered, more than 65403 students have viewed full stepbystep solutions from this chapter.

Block matrix.
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.

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

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

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.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

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.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

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

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

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.

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.

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

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

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.

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.