Polar Coordinates z=rcisӨ

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Presentation transcript:

Polar Coordinates z=rcisӨ Write z=2+3i in polar form. 1: sketch the point. 2. find modulus & argument (angle the line makes with the real axis) Modulus is √(22+32)=√13 Ө=tan-1(3/2)=0.9828rad (4dp) 2+3i=√13cos0.9828+√13isin0.983 = √13(cos0.9828+isin0.983) 3. write in polar (rcisӨ) form. = √13cis0.9828

Polar Form rcisθ=r(cos θ+isin θ) r=√ (a2+b2) r=√ (-3.22+-.92) r=3.32 90 θ 0.9 Θ=inv.tan(3.2/0.9) =74.29’ =3.2cos 3.2 Θ= --(74.29+90) Θ=--164.29’ -3.2 - 0.9i (rectangular form) Polar form is 3.32cis(-164.29’)

On GRAPHICS CALCULATOR: RUN mode->OPTN->CPLX, To find modulus: Abs(2+3i) To find argument: Arg (2+3i)

Practice: write in polar form (with arguments in radians) Z=6+i Z=-4+2i Z=-3-4i Z=2-5i Answers: a)z=6.08cis(0.1651) b)z=4.47cis(2.6779) c)z=5cis(-2.2143) c)z=5.39cis(-1.1903)

Converting from polar to rectangular form… expand out: Write z=3cis(-150°) in rectangular form. 3cis(-150)=3(cos-150+isin-150) =-2.6-1.5i Change to rectangular form: Z=4cis(27°) Z=2.3cis(140°) Z=1.9cis(-1.427rad) Z=5.4cis(-2.15rad) Ex 32.1 p.293 #2-5

Operating on Numbers in Polar Form Multiplication: multiply the moduli, add the argument. Division: divide the moduli, subtract the argument. Raising to a power (This is called DeMoivre’s Theorem) Ex 32.2 p.293 Ex 32.3 p.296 #1