Presentation on theme: "10.4 Products & Quotients of Complex Numbers in Polar Form"— Presentation transcript:
1 10.4 Products & Quotients of Complex Numbers in Polar Form
2 z1z2 = r1r2[cos(θ1 + θ2) + isin(θ1 + θ2)] Exploration!We know how to multiply complex numbers such as…(2 – 7i)(10 + 5i) = i – 70i – 35i2= i – 70i + 35 = 55 – 60iSo we can also multiply complex numbers in their polar forms.Let’s trysum identities!*this pattern will happen every time, so we can generalize it:If z1 = r1(cosθ1 + isinθ1) and z2 = r2(cosθ2 + isinθ2) are complex numbers in polar form, then their product isz1z2 = r1r2[cos(θ1 + θ2) + isin(θ1 + θ2)]product:
3 Express in rectangular form. Ex 1) Find product ofExpress in rectangular form.Draw a diagram that shows the two numbers & their product.zwzw
4 Ex 2) Interpret geometrically the multiplication of a complex number by i general polar:r (the modulus) is the samebut angle is rotated counterclockwiseEx 3) Use your calculator to find the product.
5 Electricity Application V = IZ voltage = current • impedance magnitude of voltage is real part of VEx 4) In an alternating current circuit, the current at a given time is represented by amps and impedance is represented byZ = 1 – i ohms. What is voltage across this circuit?2.73Vreal part
6 To understand the rule for dividing polar numbers, first we practice dividing rectangular complex numbers.Ex 5) Find the quotient ofRemember the multiplicative inverse of 2 is ½ since 2 • ½ = 1.The multiplicative inverse of a + bi is By getting rid of i indenominator, we would getIn general, the multiplicative inverse of z is
7 Ex 6) Find the multiplicative inverse of 5 – 14i quotient:If z1 = r1(cosθ1 + isinθ1) and z2 = r2(cosθ2 + isinθ2) arecomplex numbers in polar form, then the quotient
8 Ex 7) Find the quotientpolarpolarConvert back to rectangular.