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Polar Coordinates z= rcis Ө

Polar Coordinates z= rcis Ө. 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 √(2 2 +3 2 )=√13 Ө =tan -1 (3/2)=0.9828rad (4dp) 2+3i= √13 cos 0.9828+ √13 isin 0.983 = √13( cos 0.9828+ isin 0.983)

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Polar Coordinates z= rcis Ө

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  1. 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

  2. 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’)

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

  4. 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)

  5. 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

  6. 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

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