6 6 De Moivres Theorem and nth Roots

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6. 6 – De Moivre’s Theorem and nth Roots HW: Pg. 559 #13 -22

6. 6 – De Moivre’s Theorem and nth Roots HW: Pg. 559 #13 -22

The Complex Plane Imaginary axis a + bi bi a Real Axis

The Complex Plane Imaginary axis a + bi bi a Real Axis

Plotting Complex Numbers • Plot u = 1 + 3 i, v = 2

Plotting Complex Numbers • Plot u = 1 + 3 i, v = 2 – i, and u + v in the complex plane

Def: Absolute Value (Modulus) of a Complex Number • The absolute value or modulus

Def: Absolute Value (Modulus) of a Complex Number • The absolute value or modulus of a complex number z = a + bi is |z| = |a + bi| = √(a 2 + b 2) In the complex plane, |a + bi| is the distance of a + bi from the origin.

Trigonometric Form of Complex Numbers Imaginary axis z = a + bi r ө

Trigonometric Form of Complex Numbers Imaginary axis z = a + bi r ө Real Axis

Def: Trigonometric Form of a Complex Number • The trigonometric form of the complex

Def: Trigonometric Form of a Complex Number • The trigonometric form of the complex number z = a + bi is z = r(cosө + isinө) where a = rcosө, b = rsinө, r = √(a 2 + b 2), tan ө = b/a. The number r is the absolute value or modulus of z

Find the trig form with 0≤ө≤ 2∏ for the complex number (a) 1 -

Find the trig form with 0≤ө≤ 2∏ for the complex number (a) 1 - √(3)i (b) -3 – 4 i

Product and Quotient of Complex Numbers • Let z 1 = r 1 (cosө

Product and Quotient of Complex Numbers • Let z 1 = r 1 (cosө 1 + isinө 1) and z 2 = r 2 (cosө 2 + isinө 2). Then 1. z 1 ▪ z 2 = r 1 r 2[cos(ө 1 + ө 2) + isin(ө 1 + ө 2 )]. 2. z 1/z 2 = r 1/r 2[cos(ө 1 - ө 2) + isin(ө 1 - ө 2 )]. r 2≠ 0

Express the product of z 1 and z 2 in standard form: • Z

Express the product of z 1 and z 2 in standard form: • Z 1 = 25√ 2 (cos(-∏/4) + isin(-∏/4)) • Z 2 = 14 (cos(∏/3) + isin(∏/3))

Express the quotient z 1/z 2 in standard form: • Z 1 = 2√

Express the quotient z 1/z 2 in standard form: • Z 1 = 2√ 2 (cos(135˚) + isin(135˚)) • Z 2 = 6 (cos(300˚) + isin(300˚))

De Moivre’s Theorem • Let z = r(cosө + isinө) and let n be

De Moivre’s Theorem • Let z = r(cosө + isinө) and let n be a positive integer. Then Zn = [r(cosө + isinө)] n = rn (cosnө + isinnө)

Using De Moivre’s Theorem • Find (1 + i√ 3)3

Using De Moivre’s Theorem • Find (1 + i√ 3)3

Find [(-√ 2/2) + i(√ 2/2)]8 using De Moivre’s Theorem

Find [(-√ 2/2) + i(√ 2/2)]8 using De Moivre’s Theorem

Using your calculator:

Using your calculator:

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