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Introduction to Complex Numbers in Trigonometric Form Accelerated Precalculus •The unit imaginary number is i = √-1 . •Remember that i 2 = -1 . •A complex number is the sum of a real number and an imaginary number. •The general (standard) form of a complex number is z = a + bi , where the real number a is called the real part and the real number b is called the imaginary part of z. •The complex conjugate of a + bi is a – bi. Complex Number Operations: Let a + bi and c + di be any two complex numbers. Addition: (a + bi ) + (c + di) = ( a + c ) + (b + d)i (Add the real parts & add the imaginary) Subtraction: (a + bi ) - (c + di) = ( a - c ) + (b - d)i (Subtract the real parts & subtract the imaginary) Complex Number Operations: Multiplication: (a + bi )(c + di) (a + bi )(c + di) = ac + adi +bci + bdi2 = (ac - bd) + (ad +bc)i (Multiply like you would any binomial - FOIL ) Division: (a + bi ) / (c + di) (a + bi) (c – di) = (ac + bd) + (-ad + bc)i (c + di) (c – di) c2 + d2 (Multiply numerator & denominator by the complex conjugate of the denominator ) Geometric Representation of a complex number is graphed in the complex plane as a point (a, b). The x-axis is called the real axis and the y-axis is the Imaginary axis imaginary axis. ( -1, 2) -1 + 2i Distance from origin is called the modulus. z = a + bi = √(a2+b2) • (1, √3 ) 1 + √3 i z = 2 60° 1 Real axis ( 0, -2) • 0 - 2i Trigonometric form of a complex number (z) includes the modulus and angle formed with the positive x-axis. Convert complex numbers to their trig form The right triangle determined by z = a + bi has side lengths of a, b, and r (radius or modulus) : r = √(a2+b2) cos θ = a/r sin θ = b/r a = r cos θ b = r sin θ Therefore a + bi = (r cos θ) + (r sin θ)i or Imaginary z = r (cos θ + i sin θ) (a, b ) a+bi r b θ Real a θ is called the argument θ = tan-1(b / a) (usually in radians) Product and Quotient of Trigonometric Form Let z1= r1 (cos θ1 + i sin θ1 ) (abbreviated z1= r1cis θ1) and z2= r2 (cos θ2+ i sin θ2 ) or z2= r2 cis θ2 z1 z2 = r1 r2 [cos (θ1 + θ2 )+ i sin (θ1 + θ2)] (Multiply the moduli; add the arguments) z1 = r1 [cos (θ1 - θ2 )+ i sin (θ1 - θ2)] z2 r2 r2 ≠ 0 (Divide the moduli; subtract the arguments) Assignment Page 450-451: 1-39 odd Please remember to write down each problem, show sketches where necessary or helpful, and then show each step of the solutions. Answers only will not be accepted!