Transcript Polar Form rcis* - Math-HS

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
θ 90
3.2
-3.2 - 0.9i
(rectangular form)
0.9
•Θ=inv.tan(3.2/0.9)
=3.2cos
• =74.29’
•Θ= --(74.29+90)
•Θ=--164.29’
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)
a)
b)
c)
d)
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:
a) Z=4cis(27°)
b) Z=2.3cis(140°)
c) Z=1.9cis(-1.427rad)
d) 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