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Time series analysis 1. (a) Use Euler's identity e¡θ-cos θ + i sin θ to prove that sin θ=-(eiO , 2i (b) Use the identities above and the formula for the sum of a geometric series to prove that if n is an integer and j E 1,2,... ,n} then TL TL sin-(2Ttj/n)- n/2 so long as J关[m/2, where Laj is the greatest integer that is smaller than or equal to x (c) Show that when j 0 we have...
14 and 17 ty Recall that cos bt = (elb + e")/2 and that sin b1 = ( -e )/2i. In each of Problems 11 through 14, find the Laplace transform of the given function; a and b are real constants. Assume that the necessary elementary integration formulas extend to this case. 11. f(1) = sin bt 12. f(t) = cos bt 13. f(t) = eaf sin bt 14. f(t) = el cos bt
(5) The image of the parametrization Φ(u, u) = (a . sin(u) . cos(u), b . sin(u) . sin(e), c . cos(u)) sin(u sin() cosu with b < a, 0 r, 0 2π parametrizes an ellipsoid. u u a) Show that all the points in the image of Φ satisfy the Cartesian equation of an ellipsoid E b) Show that the image surface is regular at all points. c Write out the integral for its surface area A(E). (Do not...
Show that cos e+sin e) sec2 e+2tan 2 cos2 e (12) (a) Hence find all values of 0, where 0<0< 2T, which satisfy the equation sec20+ 2 tan 0 (b) =2(2 + tan e) (cos e+sin e)
sin^2(theta/2)/sin^2 Complete the identity. sin sin2 = ? sinde O e cos? 2 1 4- cos e O_1 2cos e O sin? 2 + 2 cos e
15. Using that sin' (2) = cos(x), cos' (2) = - sin() show that arccot (0) = 1 +22
ecos (20) cos e Establish the identity cos + cos (30) sin 0+ sin (30) cot (20) Choose the correct sequence of steps to establish the identity cos 0 + cos (30) 2 cos (20) cos (20) OA sin 0+ sin (30) cot (20) 2 cos (20) sin (20) B. cos 0 + cos (30) sin 0 + sin (30) = 2 sin (20) cos e = cot (20) Ос. = cos 0 + cos (30) 2 sin cos (20)...
1. Use the Euler identity to show that (cos(e) + isin(e)) cos(ne) + i sin(n8)
2. Let A = (cos, sin and B = (cos, sin) be two vectors on the x-y plane. Let C = (cos, sin be another non-zero vector on the x-y plane not collinear with A or B. Show that Ax B = -Bx C. If we could cancel B, as we could if these were real numbers, is it true that A= -Č? 2. Let A = (cos, sin and B = (cos, sin) be two vectors on the x-y...
Please solve this question The image of the parametrization Ф(u, u)-(a . sin(u) . cos(v), b . sin(u) . sin(v), c . cos(u)) with óくa, 0 < u < π, 0 < v < 2π parametrizes an ellipsoid. a) Show that all the points in the image of Ф satisfy the Cartesian equation of an ellipsoid E 2 b) Show that the image surface is regular at all points c) Write out the integral for its surface area A(E), (Do...