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NAME: _______________________ PRECALCULUS NOTES & EXAMPLES Section 6.5 Trigonometric Form of Complex Numbers Objectives: Plot complex numbers in the complex plane Write the trigonometric forms of complex numbers The complex number z = a + b i can be represented by a point on the complex plane. The absolute value of the complex number z = a + b i is the distance between the origin and point (a, b). Ex. Plot the complex number z = 3 – 4 i and find its absolute value. The trigonometric form of a complex number is also called the polar form. Because there are infinitely many choices for θ, the trigonometric form of a complex number is not unique. Normally, θ is restricted to the interval 0 < θ < 2 π. Represent the complex number graphically, and find the trigonometric form of the number. 1. z = 1 − 3i 2. z = −3 + ⎛ 2π Find the standard form of the complex number. z = 8 ⎜ cos 3 ⎝ 3 i + i sin 2π ⎞ 3 ⎟⎠ Multiplication and Division of Complex Numbers Suppose you are given two complex numbers z1 = r1 (cosθ1 + i sin θ1 ) and z2 = r2 (cosθ 2 + i sin θ 2 ) Find the product z1 z2 and simplify using sum and difference formulas for sine and cosine. z1 z2 = Product and Quotient of Two Complex Numbers Let z1 = r1 (cosθ1 + i sin θ1 ) z1 z2 = z1 = z2 and z2 = r2 (cosθ 2 + i sin θ 2 ) be complex numbers. Ex. Find the product of the two complex numbers. z1 = 2 (cos 2π 2π + i sin ) 3 3 z2 = 8 (cos 11π 11π + i sin ) 6 6 Ex. Find the quotient of the two complex numbers. z1 = 24 (cos 300 0 + i sin 300 0 ) z1 = 8 (cos 75 0 + i sin 75 0 ) Use the multiplication rule to find the following: z = r (cosθ + i sin θ ) z2 = z3 = z4 = . . . zn = DeMoivre’s Theorem If z = r (cos θ + i sin θ ) is a complex number and n is a positive integer, then zn = ( Ex. Use DeMoivre’s to find −1 + 3i ) 12 .