Use mathematical induction to prove the following theorem: (calculation of power

Question

Use mathematical induction to prove the following theorem:
(calculation of powers in C): Let z=r(cos ? + i sin ?) be any complex number and n be any positive integer. Then z^n=r^n(cos (n?) + i sin (n?)) or equivalently [r,?]n=[rn, n?].

Explanation / Answer

1) Since we want the lower half of the ellipsoid, we need z = 0. ==> z = sqrt(1 - 4x^2 - 2y^2) = sqrt(1 - 4u^2 - 2v^2). ------------ 2) As with 1, since x = u and y = v, z = (u^2 + v^2)^(1/2). ------------ 3) Substitute into the equation of the sphere: (8 sin f cos ?)^2 + (8 sin f sin ?)^2 + z^2 = 64 ==> 64 sin^2(f) [cos^2(?) + sin^2(?)] + z^2 = 64 ==> 64 sin^2(f) * 1 + z^2 = 64 ==> z^2 = 64 (1 - sin^2(f)) ==> z^2 = 64 cos^2(f) ==> z = 8 cos f. z = ±4 ==> 8 cos f = ±4 ==> cos f = ±1/2 ==> f = p/3, 2p/3. ------------ 4) Let R(u, v) = . Note that (x,y,z) = R(-1, 1) = (1, 1, -1). For the normal to the tangent plane: n = R_u x R_v = |.i...j...k| |2u..0..v| = |0..2v..u| At (u, v) = (-1, 1), n = . So, the equation of the tangent plane is -2(x - 1) + (-2)(y - 1) + (-4)(z - (-1)) = 0 ==> (x - 1) + (y - 1) + 2(z + 1) = 0 ==> x + y + 2z = 0.

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