1. (a) Use Euler's identity e¡θ-cos θ + i sin θ to prove that sin θ=-(eiO , 2i (b) Use the identi...
(b) Use the identities above and the formula for the sum of a geometric series to prove that if n is an integer and je[1,2,..., n) then sin2 (2ntj/n) = n/2 t-1 so long as jメ1n/21, where Ir] is the greatest integer that is smaller than or equal to x. We were unable to transcribe this image (b) Use the identities above and the formula for the sum of a geometric series to prove that if n is an integer...
2.13 Probiems 73 216 Prove de Moivre's formula (cos θ + j sin θ)" = cos(n θ) + j sin(ne). where n is an integer 217 Use de Moivre's formula, given by Eq. (2.80), to develop the rectangular and polar form representations of the (2.80) following complex numbers: 2.18 Show that 219 Determine the roots of the following second-degree polynomials (a) (G)-2s2 -4s + 10, 2.13 Probiems 73 216 Prove de Moivre's formula (cos θ + j sin θ)" =...
Problem 1: Use complex exponentials to show the following trigonometric identities: a) b) cos(4 + θ) = cos(A)cos(%)-sin(θ)sin(4) cos(0,-&J=cos(θ) cos(9a) + sin(81)sin(82).
8. (a) (5pt) Prove the identity: cscx -cos x=sec x . sin'x 4 and tan θ<0 (b) (5pt) Find cos θ , if sin θ 8. (a) (5pt) Prove the identity: cscx -cos x=sec x . sin'x 4 and tan θ
Verify that Prn (cos θ) solves sin θΟθ (sin ea, Θ) + (E(1 + 1) sin2 θ-m2) Θ 0. Use that pr(z)-(1-z2 )T (4), Pr(r) with Pr(z) a Legendre polynomi 1 Verify that Prn (cos θ) solves sin θΟθ (sin ea, Θ) + (E(1 + 1) sin2 θ-m2) Θ 0. Use that pr(z)-(1-z2 )T (4), Pr(r) with Pr(z) a Legendre polynomi 1
In questions 1-8, find the limit of the sequence. sin n cos n 2. 37 /n sin n 3. 4. cos rn 5. /n sin n o cos n n! 9. If c is a positive real number and lan) is a sequence such that for all integer n > 0, prove that limn →00 (an)/n-0. 10. If a > 0, prove that limn+ (sin n)/n 0 Theorem 6.9 Suppose that the sequence lan) is monotonic. Then ta, only if...
thx!!! previous info (iv) Explain why it follows from (iv) that IV 2T+1 I(x) = Σ 2n+1 7and (2n +1)28 Like at least one of Euler's proofs, it derives the latter first and then deduces the former from it We will work with the function sin 2θ 1 + x cos 2θ ( tan-1 where T and θ are two independent variables. Sometimes we will regard x as the variable and sometimes and we will try to keep this clear....
iv only pls (iv) Explain why it follows from (iv) that IV 2T+1 I(x) = Σ 2n+1 7and (2n +1)28 Like at least one of Euler's proofs, it derives the latter first and then deduces the former from it We will work with the function sin 2θ 1 + x cos 2θ ( tan-1 where T and θ are two independent variables. Sometimes we will regard x as the variable and sometimes and we will try to keep this clear....
5. In this problem we will derive some trigonometric identities important for deriving Fourier series. Let wo = 7, where T is a given constant. Assume k, n, m > 0 are integers. 27 (a) Show that T/2 T when k = 0 ejkwot dt =. J-T/2 O when k >1 (b) Use the above result, ej(n-m)wot = ejnwote-jmwot, ej(n+m)wot = ejnwotejmwot, and Euler's identi- ties to deduce (T/2 when n = m, n = 0, m = 0 i....
-1-1 arctan n n" n!5* (c) Find the interval of convergence and radius of convergence for )0301 i )e-3r) (d) Use the geometric series to write the power series expansion for i. f(1)- 2-4r, centered at a = 0. i.)4 centered at a-6. (e) Write the first 4 nonzero terms of the Maclaurin expansion for i, f(z) = z2 (e4-1) ii. /(x) = cos(3r)-2 sin(2x). (0) Use the Taylor Series definition to write the expansion for f(a)entered at (8) Use...