 6.3.Problem 1: Solve sin 33/2 if 0 360. Trig
 6.3.a: If is between 0 and 360, then what can you conclude about 2?
 6.3.1: To solve a trigonometric equation involving a multiple angle n, fir...
 6.3.Problem 2: Find
 6.3.b: If 230 360k, then what can you say about ? c
 6.3.2: For a trigonometric equation involving cos nin degrees, add _______...
 6.3.Problem 3:
 6.3.c: What is an extraneous solution to an equation?
 6.3.3: To solve the equation sin 3x cos x cos 3x sin x 1, use a _____ iden...
 6.3.Problem 4: Solve
 6.3.d: Why is it necessary to check solutions to equations that occur afte...
 6.3.4: True or False: To solve the equation cos 2x 3/2, use a doubleangle...
 6.3.Problem 5:
 6.3.5: sin 26.
 6.3.Problem 6:
 6.3.6: sin 2
 6.3.Problem 7:
 6.3.7: tan 2 1 8.
 6.3.8: cot 21 9
 6.3.9:
 6.3.10:
 6.3.11: cos 2x
 6.3.12: sin 2x
 6.3.13: sec 3x 1 1
 6.3.14: csc 3x 1
 6.3.15: tan 2x 3 1
 6.3.16: tan 2x 3 F
 6.3.17: sin 218
 6.3.18: sin 2
 6.3.19: cos 30 2
 6.3.20: cos 3 1 2
 6.3.21: sin 102
 6.3.22: cos 8
 6.3.23: sin 2x 24
 6.3.24: cos 2x
 6.3.25: cos 3x 2
 6.3.26: sin 3x
 6.3.27: tan 2x 2
 6.3.28: tan 2x 1
 6.3.29: sin 2x cos x cos 2x sin x
 6.3.30: sin 2x cos x cos 2x sin x
 6.3.31: cos 2x cos x sin 2x sin x
 6.3.32: cos 2x cos x sin 2x sin x
 6.3.33: sin 3x cos 2x cos 3x sin 2x 1 3
 6.3.34: sin 2x cos 3x cos 2x sin 3x 1
 6.3.35: sin2 4x 1 3
 6.3.36: cos2 4x 1
 6.3.37: cos3 5x 1 38
 6.3.38: sin3 5x 1
 6.3.39: 2 sin2 3sin 31 0 40.
 6.3.40: 2 sin2 33 sin 31 0 4
 6.3.41: 2 cos2 23 cos 21 0 42.
 6.3.42: 2 cos2 2cos 21 0 43.
 6.3.43: tan2 33 44
 6.3.44: cot2 31 F
 6.3.45: cos sin 1 46.
 6.3.46: sin cos 1 47
 6.3.47: sin cos 1 48.
 6.3.48: cos sin 1 49.
 6.3.49: sin2 24 sin 21 0 50.
 6.3.50: cos2 36 cos 34 0 51
 6.3.51: 4 cos2 38 cos 31 0 52.
 6.3.52: 2 sin2 26 sin 23 0 53
 6.3.53: 2 cos2 42 sin 41 54.
 6.3.54: 2 sin2 42 cos 41
 6.3.55: Ferris Wheel In Example 6 of Section 4.5, we found the equation tha...
 6.3.56: Ferris Wheel In of 4.5, you found the equation that gives the heigh...
 6.3.57: Geometry The following formula gives the relationship between the n...
 6.3.58: Geometry If central angle cuts off a chord of length c in a circle ...
 6.3.59: Rotating Light In Example 4 of Section 3.5, we found the equation t...
 6.3.60: Rotating Ligh
 6.3.61: sin x
 6.3.62: sin2 x (1 cos x)
 6.3.63: cos t
 6.3.64: 1 1 sin t
 6.3.65: sin 2A
 6.3.66: cos 2B
 6.3.67: cos
 6.3.68: sin
 6.3.69: Solve 2 cos 33for all degree solutions. a. 10 360k, 110 360k b. 10 ...
 6.3.70: Solve sin 4x cos x cos 4x sin x 1 for all radian solutions. a. k b....
 6.3.71: Solve 2 cos2 4cos 41 0 for all degree solutions. a. 15 30k b. 7.5 3...
 6.3.72: The height of a passenger on a Ferris wheel at any time t is given ...
Solutions for Chapter 6.3: Trigonometric Equations Involving Multiple Angles
Full solutions for Trigonometry  7th Edition
ISBN: 9781111826857
Solutions for Chapter 6.3: Trigonometric Equations Involving Multiple Angles
Get Full SolutionsThis textbook survival guide was created for the textbook: Trigonometry, edition: 7. Chapter 6.3: Trigonometric Equations Involving Multiple Angles includes 83 full stepbystep solutions. Trigonometry was written by and is associated to the ISBN: 9781111826857. This expansive textbook survival guide covers the following chapters and their solutions. Since 83 problems in chapter 6.3: Trigonometric Equations Involving Multiple Angles have been answered, more than 24586 students have viewed full stepbystep solutions from this chapter.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

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.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

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.

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

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

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

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.

Iterative method.
A sequence of steps intended to approach the desired solution.

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.

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

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

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

Solvable system Ax = b.
The right side b is in the column space of A.

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

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

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.