Prove that the second equality is true and find C and ô in terms of A, B, and w*. ylt) = e alphart(A cos w*t + B sin w*t) = Ce¯alpha*t cos(w*t – 6)

Q: A- Find all values of z such that sinh(z) = e'ő. B- Are |z], Img(z) analytic? Give reason.

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Q: 7) Use Theorem 7.1.1 (Table) to find L{f (t)} for the following: F()=cos(V2 t)+ sin(3:) n(3t) b. =…

A: Given ft=cos2t+sin3t

Q: 1. Prove that the composite trapezoidal rule with h = 2-i, i E N, is exact for ſ cos? (Tx) dr

A: We know that in composite trapezoidal rule, we approximate the integral as follows: ∫abf(x) dx ≈  h…

Q: Use symmetry to evaluate the followingintegrals.

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Q: (b) The For any X1,...X, and Y1,..., Ym, we have: n п т Cov EXi, -ΣΣCον (X, Y) i=1 j=1 i=1_j=1 Hint:…

A: From the property of covariance,  Assume that X and Y are the random variables. Therefore,  CovX,…

Q: If I, = e cos" zdr, where n > 2, Prove that i) I, =1-n e sin z cos"- rdr

A: Given,  In=∫0π2e-xcosnxdx solving by using Integration by parts, Formula of integration by parts,…

Q: -W : By the same way, prove that g Z sinWo sinw.k Z -27 cus Wo +1

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Q: 2. : (a) For n ≥ 1, show that 1+z+2²+...+ 2¹: (b) Hence, show that 1+ cos 0 + cos 20 + 12+1 2 #1 7…

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Q: •2n Let a and b be distinct nonnegative integer. Prove that Sa sinax cos bx dx = 0 %3D

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Q: Z1 "(cos(0, r2 Z2 02) + isin(0, – 02)](z2 # 0). 85. Let z1 = ri(cos 0, + isin 01) and z2 = r2(cos 0,…

A: Given z1=r1cosθ1+isinθ1 and z2=r2cosθ2+isinθ2.

Q: Let a not be an integer, and f (x) = cos ax, x E (-7, 7] Extend f to PC (27). Show that 1 2 cot aT =…

A: Using Fourier series and putting some particular value to get our required series.

Q: Q.3- Prove that: 2n 1 (cos =0 +i sin-) + 10 2cosne eino

A: The given problem is to prove the following complex identity. (cos n/5 θ +i sin 2n/10 θ)+1/einθ =…

Q: 31. Use the identity x + y y sin x + sin y = 2 sin cos 2 to prove that sin(h/2) cos h sin(a + h) –…

A: Given information: sinx+siny=2sinx+y2cosx-y2 To prove: 1: sina+h-sina=hsinh/2h/2cosa+h2

Q: Find the area lying outside r = 4 cos 0 and inside r = 2+2 cos 0.

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Q: 5. cos(M) = %3D 6. sin(L) = %3D 7. cos(L) = 30 34 %3D 8. tan(M) = %3D 16

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Q: Prove that Limit cos?P + cos²P. 2n + cos?P. 2n - p +r + 2n + cos?p_t ...... 4r I=1

A: Consider the given information. Consider the left side of the expression as,…

Q: Let z = 1- V3i and w = - v2 + vZi . Evaluate а. w 2 ( cos 12 i sin 12 O b. w -= 2 (COS 117) +i sin…

A: Given z=1-3iw=-2+2i

Q: Let z(x, y) = xy where x = r cos(80) & y = r sin( - 50). %3D | дх ду дх ду & ar' ar' a0 az dz & ar…

A: Let z = f(x,y), then,   ∂z∂r =∂z∂x · ∂x∂r + ∂z∂y · ∂y∂r   ∂z∂θ =∂z∂x · ∂x∂θ + ∂z∂y · ∂y∂θ

Q: e) cos (x + y) - cos(x - y) =-2sinxsiny

A: Given that, cosx+y-cosx-y=-2sinxsiny. Note that, the formula for cosA+B=cosAcosB-sinAsinB and…

Q: Show that (cos x + isin x)n= cos nx + isin nx whenever n is a positive integer. (Here i is the…

A: We can prove given result by mathematical induction: Define a statement p(n).  Show that p(n) is…

Q: 2.Expand the following in powers of x i)√(1 + sinx) ii)e cosx iii) cos x²+

A: Given data:- (i); 1+sinx (ii); excosx (iii); cos-1x2-1x2+1

Q: Answer the following completely. Prove: cos x + sin x sin y = sin(32) cos (2) for all x and y such…

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Q: Calculate cos(u) for the following choices of u(x) : u(x) %=D2- 2"', 옮cos(u(x) =| | u(z) %3D 2-°,…

A: Perform the differentiation

Q: (B)JE w = x*+y+z° and x = PSin ø cos@ »y=pSing Sino and z- Pcoseo prove that -2P Cos 2Ø and' %3D

A: Given  w=x2+y2-z2x=ρsinϕcosθy=ρsinϕsinθz=ρcosϕ

Q: Prove thet (1+ cose -isine)" +(l+ Cos e+isine)" = 2""cos (em). %3D Co."(을) COS

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Q: Evaluate the integral (@) = / cos (0) + sin(20) – 3 cos(0) sin (0) + 3 cos2(0) + 2 sin(0) COS de 3…

A: Evaluate the integral

Q: Assume cos(t) = - where <t <a. Compute the following %3D cos(-t) = sin(t) = sin(-t) =U sin(-t) +…

A: topic - trigonometric ratios sin(-x)=-sin(x) cos(-x)=cos(x) sin2(x)+cox2(x)=1

Q: S-) Find the gereral solufon of Sin'x y"- (2sinx.cos x)g'+ Ccosx+1)y = Sih? x given that y-sihx and…

A: Given differential equation is sin2xy"-2sinxcosxy'+cos2x+1y=sin3x Here, qx=sin3x Given that y1=sinx…

Q: 2) If z = cos 0 +isin 0 show that 1 = 2 cos 0 a) z+-D 1 b) z" + 2 cos no %3D

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Q: Evaluate the iterated integral. π/4 cos(8) 1¹ Jo 3/² 3r² sin(8) dr de

A: Given: ∫0π4∫0cosθ3r2sinθdrdθ

Q: Let = r, (cos 0, + isin 0, ) and r, (cos 0, + i sin 0, ). Z, = X1 + iy, 1 Z 2 X2 + iy, Then show…

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Q: Prove that the following equation is an identity. Hint: In = In (a) - In (b) and In (ab) = In (a) +…

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Q: Calculate cos(u) for the following choices of u(x) : dx u(x) = 7 – x², cos(u(x)) = %3| dæ u(x) =…

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Q: Show that J, (z cos0)d0 = - R/2 os e)de = 1-cos z %3D 21

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Q: Let A E R²x2 be special orthogonal. Show that there exists & E [0,27t) such that COS O - sin o A =…

A: According to question,Given Matrix,A = cos ϑ-sin ϑsin ϑcos ϑ

Q: Given that cos(2t) = 2 cos (t) – 1, it follows that L{cos"(t)} =;( +:). O True O False

A: We need to find Laplace Transform of f(t) = cos2(t) But , cos2t = 2cos2(t) - 1  ⇒ 2cos2(t) = 1 +…

Q: Express Q(x) = cos4 (x )+ 12 as the composition of three func-tions; that is, identify f, g, and h…

A: Q(x) = cos4 (x )+ 12 We need to express the given function as a composition of three functions  that…

Q: Let w = x*y cos(42). Find the following. (a) (x, y, z) = %3D (b) (x, y, z) = ду (c) (x, y, z) = dz

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Q: Prove the following idenbter P Sin (D + Sin(T a Cos A a 4

A: Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: Find the exact values of s in the given interval that satisfy the given condition. [0, 2n); cos s=-

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Q: b. Determine real (e/z) in terms of x and y. C. Use Euler's formula to prove that expression; = cos…

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Q: (a) Evaluate O ezz C cos(TZ) where c is z = 1

A: The objective is to evaluate ∮Cez2cosπz where C is z=1 Singularity of ez2cosπz are,…

Q: If the coordinates of any two points ₁ and ₂ are (x₁, y₁) and (x2, ₂), respectively, then prove that…

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Q: 3. Prove there is at least one x E R such that e* 2 cos x +1

A: We have to show there exist x€R such that e^x=2cosx+1...... (I)  We plot the graph of both function…

Q: Prove that the inequality | cos(r)| 2>1- sin°(x) holds true for all r e R.

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Q: (d) Show that there is some x with 0 < x < 2 such that x2 + cos(ax) = 4. %3D

A: Solution-