math135-s11-2011

math135-s11-2011 - Assignment 11 Solutions 1. Express the...

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Unformatted text preview: Assignment 11 Solutions 1. Express the complex number (1- 3 i ) 8 in standard form. Solution: | 1- 3 i | = 2 and the argument of (1- 3 i ) is = 5 / 3. So (1- 3 i ) = 2(cos(5 / 3)+ i sin(5 / 3)). By De Moivres Theorem, we have (1- 3 i ) 8 = 2 8 (cos(40 / 3) + i sin(40 / 3)) = 2 8 (cos(4 / 3) + i sin(4 / 3)) = 128(- 1- 3 i ) 2. (a) Use De Moivres Theorem to prove that cos4 = 8cos 4 - 8cos 2 + 1 sin4 = 4cos (sin - 2sin 3 ) . Solution: Using De Moivres Theorem, cos4 + i sin4 = (cos + i sin ) 4 and by the Binomial theorem, (cos + i sin ) 4 = cos 4 + 4 i cos 3 sin - 6cos 2 sin 2 - 4 i cos sin 3 + sin 4 . Hence, cos4 = cos 4 - 6cos 2 sin 2 +sin 4 sin4 = 4cos 3 sin - 4 i cos sin 3 = 4cos (cos 2 sin - sin 2 ) Using sin 2 = 1- cos 2 we obtain cos4 = 8cos 4 - 8cos 2 + 1 sin4 = 4cos (sin - 2sin 3 ) ....
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math135-s11-2011 - Assignment 11 Solutions 1. Express the...

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