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PreCalculus Class Notes PC4 Trigonometric Form and Roots of Complex Numbers Graphing complex numbers on the Complex Plane (rectangular coordinates): the horizontal axis (x) is the real axis and the vertical axis (yi) is the imaginary axis. Examples: Graph and label A 2 B 3i C –4 D –2i E 1–i F –2 +3i Trigonometric Form A complex number is expressed in standard form as a + bi where a is the real part and b is the imaginary part. Trig form of a complex number is similar to polar coordinates (but is not a coordinate). The expression r ( cos θ + i sin θ ) , abbreviated rcisθ , is called the trigonometric form of the complex number a + bi, where a = r cos θ and b = r sin θ. The number r is the modulus (magnitude) of a + bi, and θ is the argument (angle) of a + bi. Example Find the radius and angle (nearest tenth of a degree) for the point 2 + 3i. Write the complex number in trig form. From standard form a + bi to trig form r ( cos θ + i sin θ ) r, magnitude (modulus), distance from the origin θ, argument, angle from the positive x-axis r = z = a2 + b2 θ = tan −1 a b From trig form r ( cos θ + i sin θ ) to standard form a + bi a = r cos θ b = r sin θ Example Write the complex number as a + bi, where a and b are real numbers. π π 4 cos + i sin 3 ( cos150° + i sin150° ) 2 2 Example Find the trigonometric form for each complex number, where 0º ≤ θ ≤ 360º. 1+i −1 − i 3 Products and Quotients of Complex Numbers Multiply two complex numbers in std form z1 ⋅ z2 = ( a1 + b1i )( a2 + b2i ) Distribute and combine Divide two complex numbers in std form z1 ( a1 + b1i ) ( a2 − b2i ) = ⋅ z2 ( a2 + b2i ) ( a2 − b2i ) Multiply numerator and denominator by conjugate of denominator z1 = r1 ( cos θ1 + i sin θ1 ) and z2 = r2 ( cos θ 2 + i sin θ 2 ) Multiply two complex numbers in trig form z1 ⋅ z2 = r1 ⋅ r2 ( cos (θ1 + θ 2 ) + i sin (θ1 + θ 2 ) ) Multiply r, add angles Divide two complex numbers in trig form z1 r1 = ( cos (θ1 − θ 2 ) + i sin (θ1 − θ 2 ) ) z2 r2 Divide r, subtract angles Example Find the product and quotient of z1 = 4(cos 45º + i sin 45º) and z2 = 2(cos 135º + i sin 135º) Product z1 ⋅ z2 = r1 ⋅ r2 ( cos (θ1 + θ 2 ) + i sin (θ1 + θ 2 ) ) Quotient z1 r1 = ( cos (θ1 − θ 2 ) + i sin (θ1 − θ 2 ) ) z2 r2 Powers of complex numbers Standard form, ( a + bi ) n Either distribute and combine OR Pascal’s triangle, powers of terms, simplify powers of i, combine like terms Trig form use De Moivre’s Theorem z n = r n ( cos ( nθ ) + i sin ( nθ ) ) power on r, multiply angle by the value of the power Example 3 Use De Moivre’s theorem to evaluate ( 2cisπ ) a. Write answer in trig form b. Write answer in standard form Example Use De Moivre’s theorem to evaluate (1 + i)8 and express the result in standard form. a. Write 1 + i in trig form b. Apply De Moivre’s Theorem c. Convert complex number from trig form to standard form