Chap 1 The Number System

65 we have cos 3 i sin 3 cos i sin 3 cos3 3i

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Unformatted text preview: at √ zw = 2 2 cos Note also that √ √ √ zw = (1 + i)(−1 − i 3) = ( 3 − 1) − i( 3 + 1), so that cos π 2π − 4 3 √ = 3−1 √ 22 and sin π 2π − 4 3 √ =− 3+1 √. 22 π 2π − 4 3 + i sin π 2π − 4 3 √ 5π = 2 2 cos − 12 + i sin − 5π 12 . On the other hand, it follows from (10) that √ z 2 = w 2 cos π 2π + 4 3 + i sin π 2π + 4 3 1 =√ 2 cos 11π 11π + i sin 12 12 . Example 1.6.4. Suppose that z = 1 + i. Then repeated application of (9) yields √ 5π 5π z 5 = 4 2 cos + i sin 4 4 √ 3π = 4 2 cos − 4 + i sin − 3π 4 . Note that we have to subtract 2π to get the principal argument of z 5 . Chapter 1 : The Number System page 11 of 19 First Year Calculus c W W L Chen, 1982, 2005 Our last example suggests the following important result. PROPOSITION 1E. (DE MOIVRE’S THEOREM) Suppose that n ∈ N and θ ∈ R. Then cos nθ + i sin nθ = (cos θ + i sin θ)n . Proof. This follows from repeated application of the product formula in polar coordinates to...
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This note was uploaded on 02/01/2009 for the course MATH 2343124 taught by Professor Staff during the Fall '08 term at UCSD.

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