s1_95-100a

s1_95-100a - cos(3 ) = cos 3 ( )-3 cos( ) sin 2 ( ) . 6....

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ACM95a/100a September 30, 2009 Problem Set I 1. Show that, for a real number n , each of the two numbers z = 1 ± ni satisfies the equation z 2 - 2 z + 1 + n 2 = 0. 2. Write the following complex numbers in the form a + ib (a) ± 2 + i 6 i - (1 - 2 i ) ² 2 (b) (1 - i ) 7 3. Find all values of (a) ( - 1) - 1 / 4 (b) 16 1 / 8 and show them graphically. 4. Sketch the regions or curves described by (a) | z - 1 + i | ≤ 1; (b) | z - i | = | z + i | ; (c) < ( z ) - = ( z ) = 5; (d) | z - i | + | z + i | = 1 . 5. Use de Moivre’s formula to derive the trigonometric identity
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Unformatted text preview: cos(3 ) = cos 3 ( )-3 cos( ) sin 2 ( ) . 6. Solve the equation | e i-1 | = 2 for (0 ) and verify the solution geometrically. 7. Show, both by geometric and algebraic arguments, that for complex numbers z 1 and z 2 the inequalities || z 1 | - | z 2 || | z 1 + z 2 | | z 1 | + | z 2 | hold. Due at 5pm on Wed. October 7....
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This note was uploaded on 01/14/2010 for the course ACM 1 taught by Professor Prof during the Fall '09 term at Caltech.

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