Presentation on theme: "10.4 Products & Quotients of Complex Numbers in Polar Form"— Presentation transcript:
110.4 Products & Quotients of Complex Numbers in Polar Form
2z1z2 = 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:
3Express in rectangular form. Ex 1) Find product ofExpress in rectangular form.Draw a diagram that shows the two numbers & their product.zwzw
4Ex 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.
5Electricity 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
6To 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
7Ex 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
8Ex 7) Find the quotientpolarpolarConvert back to rectangular.