Given that z= 1+i√2, express in the form a+bi each of the complex numbers p=z+1/z, q=z-1/z. In an Argand diagram, P and Q are the points which represent p and q respectively, O is the origin, M is the midpoints of PQ and G is the point on OM such that OG=(2/3)OM. Prove that angle PGQ is a right angle.
(1) Express -1+i in polar form. Hence show that (-1+i)^16 is real and that 1/(-1+i)^6 is purely imaginary, giving the value for each.
(2) Express sin5α and cos5α/cosα in terms of sinα
Simplify the following expressions: (a) (cos π/4 + i sin π/4)(cos 3π/4 + i sin 3π/4), (b)
√ 3, (b) w = −2 1 √
(cos π/4 + i sin π/4)2 (cos π/6 + i sin π/6)
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