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• ### Recall that, for all c, = n=0 cos(x) = § 4 (-1)",21 (2n)! and sin(x) =...

Recall that, for all c, = n=0 cos(x) = § 4 (-1)",21 (2n)! and sin(x) = (-1)"..2n+1 (2n + 1)! N=0 n=0 If i is defined to have the property that i = -1, show that ei cos(2) + isin(x) for any real number r.

• ### Show that the wafunctions sin "/C and cos "2* are orthogonal over the interval 0sxsa. n...

Show that the wafunctions sin "/C and cos "2* are orthogonal over the interval 0sxsa. n is an integer and cOS_ are

• ### For the set of functions {sin(x),sin(2x),sin(3x),...}=sin(nx)}, n=1,2,3,... on the interval [0,pi...

For the set of functions {sin(x),sin(2x),sin(3x),...}=sin(nx)}, n=1,2,3,... on the interval [0,pi]. Show that the set of functions is orthogonal on [0,pi].

• ### Establish the identity. 1 - sin 0 cos e + COS 0 1 - sin e...

Establish the identity. 1 - sin 0 cos e + COS 0 1 - sin e = 2 sec Write the left side of the expression with a common denominator. Do not expand the numerator. cos (1 - sin o) Expand and simplify the numerator by rewriting without any parentheses. + cos20 cos (1 - sin o) Apply an appropriate Pythagorean Identity to simplify the numerator of the expression from the previous step. cos (1 - sin o) (Do not...

• ### n=0 4. Using the power series cos(x) = { (-1)",2 (-0<x<0), to find a power (2n)!...

n=0 4. Using the power series cos(x) = { (-1)",2 (-0<x<0), to find a power (2n)! series for the function f(x) = sin(x) sin(3x) and its interval of convergence. 23 Find the power series representation for the function f(2) and its interval (3x - 2) of convergence. 5. +

• ### Let Xn = a sin(bn+Z), where n ∈ Z, a, b ∈ [0, ∞) are constant, and Z has a continuous uniform distribution on [−π, π] (i...

Let Xn = a sin(bn+Z), where n ∈ Z, a, b ∈ [0, ∞) are constant, and Z has a continuous uniform distribution on [−π, π] (i.e. Z ∼ U([−π, π])). Show that Xn is stationary. (Hint: sin(x) sin(y) = 1 2 (cos(x − y) − cos(x + y)) may be helpful). l. Let Xn-a sin(bn+ Z), where n є z, a, b є lo,00) are constant, and Z has a continuous uniform distribution on [-π, π] (i.e. Z ~...

• ### Question 9 Find all solutions to the equation in the interval [0, 2n). sin 2x -...

Question 9 Find all solutions to the equation in the interval [0, 2n). sin 2x - sin 4x = 0 Your answer: O O 51 71 I, 31 111 z' ' ä 'ö' ' ő 31 1171 Oo, ma Clear answer Question 10 Find all solutions to the equation in the interval [0, 21). cos 4x - cos 2x = 0 Your answer: o o, 110 TT 51 71 31 6' 2' O No solution Clear answer Question 11 Rewrite...

• ### For each of the following functions indicate the matching Taylor Series centered at r=0. 1) sin(2)...

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• ### n, fx/<1/2n 5. In the interval (-17, T), O, (x) = jo, x]>1/2n (a) Expand 8,...

n, fx/<1/2n 5. In the interval (-17, T), O, (x) = jo, x]>1/2n (a) Expand 8, (x) as a Fourier cosine series. (b) Show that your Fourier series agree with a Fourier expansion of d(x) in the limit as n →00.

• ### Establish the identity 1 - cos 0 sin 0 + sin 0 1 - cos 0...

Establish the identity 1 - cos 0 sin 0 + sin 0 1 - cos 0 = 2 csc 0. Which of the following shows the key steps in establishing the identity? 1 - cos e sin 0 ОА. + sin e 1 cos e 1 - cos e B + sin e 1 - cos 0 sin e (1 - cos 0)2 + sine 2 = 2 csc 6 sin 0(1 - cos ) cOS (1 - cos 02...