Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Section 6.5 Complex Numbers in Polar Form Overview • Recall that a complex number is written in the form a + bi, where a and b are real numbers and i 1 • While it is not possible to graph complex numbers on a real number plane, a similar setup can be used. The Complex Plane Graph Each of the Following • z = 3i • z = -5 + 2i • z = 3 – 4i Absolute value of a complex number • The absolute value of a complex number z is the distance from the origin to the point z in the complex plane: z a b 2 2 Polar form of a complex number • When a complex number is in a + bi form, it is said to be in rectangular form. • Just as we superimposed the polar plane onto the rectangular coordinate plane, we can do the same thing with the complex plane. Continued z r (cos i sin ) a r cos b r sin r a2 b2 b tan ,0 2 a •r is called the modulus and “theta” is called the argument. Examples • Graph each of the following, then write the complex number in polar form: 1 i 5 3 3 3i Now, the Other Way • Write each complex number in rectangular form: 3cos 330 i sin 330 4 cos i sin 4 4 Products and Quotients • Given z1 r1 cos 1 i sin 1 z2 r2 cos 2 i sin 2 , two complex numbers in polar form. • Their product and quotient can be found by the following: z1 z 2 r1r2 cos1 2 i sin 1 2 z1 r1 cos1 2 i sin 1 2 z 2 r2 In Other Words… • When multiplying, multiply the moduli and add the arguments. • When dividing, divide the moduli and subtract the arguments. • Keep in mind that you may have to rename your argument so that is an angle between 0 and 360° or 0 and 2π radians. Raising to a Power When raising a complex number to a power, use DeMoivre’s Theorem: n n z r cos n i sin n In other words, raise the modulus to the nth power and multiply the argument by n (again, be prepared to rename your argument). A Final Word Before the Examples • Pay particular attention to the form your final answer should take (complex polar or complex rectangular). Find the Product (Answer in Polar Form) z1 cos z 2 cos 4 6 z1 2 2i z 2 1 i i sin i sin 4 6 Find the Quotient z1/z2(Answer in Polar Form) z1 72cos 12 i sin 12 z 2 9cos 4 i sin 4 z1 4 cos i sin 10 10 z 2 9 cos i sin 12 12 Use the French Guy’s Theorem (write answers in rectangular form) 6cos15 i sin 15 3 2 2 3 cos i sin 3 3 8 Finding Complex Roots Let w = r(cos θ + i sin θ) be a complex number in polar form. w has n distinct complex nth roots given by 2k 2k zk r cos i sin , k 0,1,2,..., n 1 n n 360k 360k zk n r cos i sin , k 0,1,2,..., n 1 n n n Examples • Find all the complex cube roots of 8. Write your answers in rectangular form. • Find all the complex fourth roots of 16(cos120° + I sin120°). Write your answers in polar form.