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Some Solutions for 6.5 Trigonometric Form of a Complex Number 1. Find the 8 roots of unity, that is, solve x8=1 roots are: 1, √2 2 + √2 𝑖 2 , 𝑖, −√2 2 + √2 𝑖, 2 -1, −√2 √2 − 2 𝑖, 2 −𝑖, √2 2 − √2 𝑖 2 2. Prove |z1z2| = |z1|*|z2| |(𝑎 + 𝑏𝑖)(𝑐 + 𝑑𝑖)| = |(𝑎𝑐 − 𝑏𝑑) + (𝑎𝑑 + 𝑏𝑐)𝑖| = √(𝑎𝑐 − 𝑏𝑐)2 + (𝑎𝑑 + 𝑏𝑐)2 = √𝑎2 𝑐 2 + 𝑏 2 𝑑2 + 𝑎2 𝑑2 + 𝑏 2 𝑐 2 = √𝑎2 (𝑐 2 + 𝑑2 ) + 𝑏 2 (𝑐 2 + 𝑑2 ) = √(𝑎2 + 𝑏 2 )(𝑐 2 + 𝑑2 ) = √(𝑎2 + 𝑏 2 ) · √(𝑐 2 + 𝑑2 ) = |z1|*|z2| 3. One can use the quadratic formula z =( -b + (b2-4ac)1/2)/2a where a, b and c are the complex numbers. Both square roots are to be considered when discriminant is non-zero. - find the roots of the equation z2 + 2z + (1 – i) = 0 - Solutions: −1 + - find the roots of the equation z2 – 4z + (4 + 2i) = 0 Solutions: 3 − 𝑖 and 1 + 𝑖 √2 2 + √2 𝑖 2 and −1 − √2 2 − √2 𝑖 2 4. Use properties of conjugates and moduli to show that a) z 3i z 3i b) iz iz c) (2 i ) 2 3 4i d) (2 z 5)( 2 i ) 3 2 z 5 hint: use property from #2 on this w.s. 5. Prove that a) Z is real if and only if z z b) Z is either real or pure imaginary if and only if ( z ) 2 z 2