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Unformatted text preview: Prove that NO order relation can be placed on C so that C becomes an ordered ﬁeld. 7. Let r , s ≥ 0, θ , φ ∈ R . Let z 1 = r (cos θ + i sin θ ) , z 2 = s (cos φ + i sin φ ) . Prove that z 1 z 2 = rs (cos( θ + φ ) + i sin( θ + φ )) . 8. Determine all solutions z in C of the equation z 3 + 64 i = 0 . Write the solutions in rectangular form and simplify. Exhibit the solutions geomet-rically. 9. Let θ ∈ R . (a) Prove that cos θ = 1 2 ( e iθ + e-iθ ). (b) Prove that sin θ = 1 2 i ( e iθ-e-iθ ). (c) Expand ( e iθ + e-iθ ) 3 and simplify. (d) Hence show that cos 3 θ = 1 4 cos3 θ + 3 4 cos θ....
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- Spring '08
- coherent, Prime number, Multiplicative inverse, Inverse element