assignment5 - ma2176 a5 Complex Numbers 1 2 Find the square...

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1 ma2176 a5 Complex Numbers 1. Find the square roots of 5 12 i . 2. Simplify ( ) 5 1 3 i − + . 3. Find the modulus and argument of ( ) ( ) 2 1 1 3 1 i i i + − + − + . 4. If ω ( ) 1 is a complex cube root of unity, show that (a) ( )( ) 2 2 1 1 4 ω ω ω ω + + = . (b) ( ) ( ) 6 6 2 2 2 5 2 2 2 5 729 ω ω ω ω + + = + + = . 5. Find the modulus and argument of 1 cos sin 1 cos sin i i θ θ θ θ + + when 0 θ π < < . What are the modulus and argument when 2 π θ π < < ? 6. Solve the equation 4 1 3 z i + = in the set of all complex numbers. 7. Solve the equation 6 4 2 1 0 z z z + + + = . 8. Show that (a) 2 4 cos5 1 12sin 16sin cos θ θ θ θ = + . (b) 4 3 64sin cos cos7 cos5 3cos3 3cos θ θ θ
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