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Section 5.3 – The Complex Plane; De Moivre’s Theorem

Section 5.3 – The Complex Plane; De Moivre’s Theorem. Complex Plane. Imaginary Axis. y. Real Axis. O. x. Cartesian Form. Polar Form. Plot 3 – 4i in the complex plane and write it in polar form. Express the argument in degrees. 3. θ. real axis. α. z = 3 – 4i. -4. imaginary axis.

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Section 5.3 – The Complex Plane; De Moivre’s Theorem

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  1. Section 5.3 – The Complex Plane; De Moivre’s Theorem

  2. Complex Plane Imaginary Axis y Real Axis O x

  3. Cartesian Form Polar Form

  4. Plot 3 – 4i in the complex plane and write it in polar form. Express the argument in degrees.

  5. 3 θ real axis α z = 3 – 4i -4 imaginaryaxis z = x +yi z = rcosθ +(rsinθ)i z = r(cosθ +isinθ) z = 5(cos 306.87o +isin 306.87o)

  6. Write the point in rectangular form r = 3 θ = 3π/2 x = r cosθ y = r sinθ y = 3 sin(3π/2) x = 3 cos(3π/2) x = 3(0) = 0 y = 3(-1) = 3 z = x + yi z = 0 - 3i z = - 3i

  7. Theorem

  8. If z = 3(cos130o + isin130o) and w = 4(cos270o + isin270o), what is zw? r1= 3, r2= 4 θ1 = 130o, θ2 = 270o zw = r1r2[cos(θ1 + θ2) + isin(θ1 + θ2)] zw = (3)(4)[cos(130o + 270o) + isin(130o+ 270o)] zw = 12[cos(400o) + isin(400o)] 40o and 400o are coterminal zw = 12[cos(40o) + isin(40o)]

  9. Theorem DeMoivre’s Theorem

  10. Write the expression in standard form a + bi = z6

  11. r = 27, θ = 60o

  12. Theorem Finding Complex Roots

  13. θ α Find the complex fourth roots of

  14. Change 2π to 360o since our angle is in degrees, set n = 4 since we are finding the complex fourth roots, plug in r = 2 and θ = 120o

  15. We can simplify the fractions

  16. Use values of k from 0 to n-1, that is 0 to 3

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