WEBVTT mathematics/trigonometry/murray
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Hi, these are the trigonometry lectures on educator.com and today we're going to talk about polar form of complex numbers.
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A lot of what we're learning in this lecture is very directly related to polar coordinates.
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If you're a little rusty on polar coordinates, what you might want to do is go back and review what you learned about polar coordinates before we learn about polar forms of complex numbers.
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In particular, the main formulas for converting a complex number into polar form, they're exactly the same formulas that you learned for polar coordinates.
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They should be familiar to you when we go through them now.
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If they're very rusty, you might want to go back and practice those formulas for converting a point into polar coordinates and back, because they'll be really helpful in this section of polar forms of complex numbers.
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Let's start out there.
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Complex numbers can be written in rectangular form, z=x+yi.
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That represents, if you graph it, then you have an x-coordinate and a y-coordinate.
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We write the rectangular formula complex number as x+yi.
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Just like with points, you would give the coordinates as (x,y), with complex numbers, we give the form as (x+yi).
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They can also be written in polar form, z=re^iθ.
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That represents the polar coordinates of the same point.
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re^iθ, sometimes people write it as re^θi.
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That represents the polar coordinates of the point, r is the radius from the origin going diagonally instead of going in a rectangular fashion.
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θ represents the angle that makes with the positive x-axis.
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Just like we've had polar coordinates rθ who have the polar form of a complex number re^iθ.
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The conversion's back and forth between those two forms are exactly the same as what we've had for polar coordinates.
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Let's check those out.
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The conversion for r is square root of x²+y².
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That comes straight from the Pythagorean theorem.
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The conversion for θ is a little more complicated and it's got the same kind of subtleties and nuances that it had with polar coordinates.
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θ is either arctan(y/x) or π+arctan(y/x).
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The way you know which one of these formulas to use is you check the sine of x.
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This is when x is greater than 0.
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This is when x is less than 0.
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Another way to remember that is to ask whether the point is in quadrant 1, 2, 3, or 4.
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Remember arctangent will always give you a value in quadrants 1 or 4.
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If you start out in quadrant 1 or 4, then you just want to use the arctangent function directly.
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If you're looking for point in quadrants 2 or 3, then the arctangent will not give you the right value, that's why you add π to it.
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That's the tricky one.
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x and y, same formulas as we had for polar coordinates before, rcos(θ) and rsin(θ).
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We'll try to use values of r that are positive, but that's not absolutely essential.
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We'll try to use values of θ that are between 0 and 2π, but that's not absolutely essential.
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Let me give you one more formula that's very very useful in working out conversions between rectangular and polar coordinates.
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We write re^iθ as x+yi.
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I'll write that as iy.
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Polar form is re^iθ, rectangular form is x+iy.
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If you convert that, the x is rcosθ, iy is irsin(θ).
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If you factor out an r there, we get r×cos(θ)+isin(θ).
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If you just take r=1, if you factor out the r from both sides, what you get here is the e^iθ=cos(θ)+isin(θ).
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That is an extremely useful formula in converting complex numbers to polar coordinates.
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That one is probably worth memorizing as well.
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c=cos(θ)+isin(θ).
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Let me decorate that a little bit, illustrate how important it is.
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e^iθ=cos(θ)+isin(θ), that's definitely worth remembering.
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We'll be using it on some of the examples.
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Let's go ahead and practice doing some conversions here.
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One more thing that I need to show you before we practice that.
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Multiplying two complex numbers in polar form.
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If we have two complex numbers in polar form, r₁×e₁^iθ, it's got an r and a θ, and r₂×e₂^iθ.
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There's a very easy way to multiply them.
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If we multiply these together, what we do is we just multiply the r's together r₁×r₂.
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Remember the laws of exponents x^a×x^b=x^a+b.
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Here, we have x or e^iθ₁×e^iθ₂, you add the exponents, iθ₁+iθ₁, just gives you i(θ₁)+θ₂).
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You add the exponents there.
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You end up just multiplying the r's and adding the angles θ₁+θ₂ because they're in the exponents.
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Now let's try some examples.
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We're going to convert the following complex numbers from rectangular form to polar form.
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Let's start out with -3+i.
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The -3 is x and y is 1 there.
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We want to find the r and θ, r is the square root of x²+y².
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Let me write this at the top page so I don't have to keep rewriting it.
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θ=arctan(y/x), that's if x > 0, or we might have to add π to that if x < 0.
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In this case, our r is the square root of x², negative root 3 squared is just 3, +y² is 1, that simplifies down to 2.
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θ=arctan(y/x), y=1, 1 over negative root 3, which is arctan of negative root 3 over 3.
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That's one of my common values.
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I know what the arctan negative root 3 over 3 is, it's -π/6.
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My x-coordinate was negative there so I haven't actually been using the right formula, I have to add π to each of these, +π.
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You almost always use radians and not degrees here.
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If you do happen to plug this into your calculator, make sure your calculator is in radian mode.
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I didn't have to use my calculator on this one because negative root 3 over 3 is a common value.
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(-π/6)+π=5π/6.
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My polar form for that complex number is r₂e^iθ, so e^(5π/6)i.
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Let's keep going with the next one.
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6 root 2, that's my x, -6 root 2, that's my y, r is the square root of x², 6 root 2 squared is 36, times 2 is 72, +y² is 6 root 2 again, 72, square root 144 is 12.
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θ=arctan(y/x), that's negative 6 root 2 over 6 root 2, which is arctan(-1) which is -π/4.
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My x in this case was positive so I don't have to introduce that correction term.
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I get w=re^iθ=12e^(-π/4).
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I don't really like that negative value of π/4, so what I'm going to do is to make it positive, to get it into the range, 0 to 2π, I'll add 2π to it.
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I'll write that as 12e, I need an i there, ex^(7π/4)i.
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You can also understand these things graphically.
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Let me draw a unit circle here.
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Negative root 3 plus i, that means my x is negative root 3 and my y is 1.
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I recognize that as a multiple of root 3 over 2 and -1.
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I recognize that as being over here.
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That's z with radius of 2, because it's 2 times root 3 over 2 and 1/2, I know that that's 5π/6.
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That's the way to kind of check graphically that my z is 2×e^(5π/6)i.
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For w, 6 root 2 minus 6 root 2, I know that's 12 times root 2 over 2 root 2 over 2, except the y is negative.
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That value is 7π/4, that's kind of a little graphical check that we have the right polar form for the complex numbers.
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Let's go back and recap what we did for that problem.
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We're converting complex numbers from rectangular form to polar form, really just boils down to these two conversion formulas for r and θ.
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r gives you the magnitude, θ gives you the angle.
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The problem though is that this θ formula is little bit tricky.
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It has this two cases depending on whether x is positive or negative.
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If x is negative then you have to add an extra π to it, that's what we did here, we were adding an extra π to the value of θ.
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Once you find r and θ, you just plug them into this form re^iθ.
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That's how we got the answers for each of those.
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For the next one, we're converting from polar form to rectangular form.
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We're given z=4e^(-2π/3)i, and w=2e^(3π/4)i.
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Let me write down the conversion formulas.
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x=rcos(θ), y=rsin(θ).
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For the first one, x=4cos(-2π/3).
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Let me graph that quickly on the unit circle.
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-2π/3 is down here, it's the same as 4π/3.
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The cosine is -1/2, that's a common value.
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This is 4×-1/2, which is -2.
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The y there is 4sin(-2π/3), the sine of that is negative root 3 over 2.
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This is 4 times negative root 3 over 2, which is -2 root 3.
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We're going for the form x+yi, our z is equal to x=-2, +yi, -2 root 3, i.
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For the second one, we have 2e^(3π/4)i.
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I'll graph that on the unit circle to help me find the sine and cosine.
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3π/4 is over there, it's 45-degree angle on the left-hand side.
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I know the sine and cosine very quickly.
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x is equal to r which is 2, cosine of 3π/4, which is 2, cosine of that is negative because it's on the left-hand side.
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2 times negative root 2 over 2, which is just negative root 2.
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y=2sin(3π/4), which is 2 times positive root 2 over 2 because we're in that second quadrant, y-coordinate is positive.
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x+yi is negative root 2 plus root 2i.
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That one wasn't too bad, it was simply a matter of remembering x=rcos(θ), y=rsin(θ), then putting those into x+yi.
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For finding the sines and cosines, it helps if you graph the angle in each case.
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Once you remember those formulas, you just work in through arccos(θ) and arcsin(θ) in each case.
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For the third example, we're going to use polar form in an application.
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We're going to perform a multiplication by converting each one of the complex numbers to polar form, then we're going to check the answer by multiplying them directly in rectangular form.
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-1 plus root 3i, I'm going to figure out my r there.
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My r is equal to square root of x²+y².
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Let me write these formulas generically, x²+y², θ=arctan(y/x), that's if x is bigger than 0, we'll have to add on a π, the fudge factor π if x < 0.
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In the first one, r is equal to 1² plus root 3 squared, that's 3, which is 2 square root of 1 plus 3.
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θ is equal to arctan negative root 3 over 1.
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Let me write that as root 3 over -1.
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I have to add on a π because the x is negative.
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Arctan of negative root 3 is negative π/3+π, that was a common value that I remembered there.
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Plus π gives me 2π/3.
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That tells me my r and my θ for the first one.
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Let me go ahead and figure them out for the second one before I plug them in.
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For the second one, we have r is equal to the square root of 2 root 3 squared.
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2 root 3 squared is, 4 times 3, is 12, plus 2 squared is 4, 12+4=16, that gives me root 16 is 4.
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θ is arctan 2 over 2 root 3, but the x-coordinate was negative, I have to add a π, so this is arctan 1 over root 3, is root 3 over 3 plus π.
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Again, that's a common value, so arctan of root 3 over 3, I remember that's a common value, that's (π/6)+π=7π/6.
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If I convert one to each one of these numbers into polar form, this one is 2e^(2π/3)i.
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This one is 4e^(7π/6)i.
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I want to multiply those, but multiplying numbers in polar form is very easy.
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First, you multiply one radius by the other one, that's 2×4=8, then you add the angles e^((2π/3)+(7π/6))i.
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You just add the angles, you multiply their radius by the other one and then you add the angles.
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That's 8e to the, let's see (2π/3)=4π/6, you get (11π/6)i.
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I want to convert that back into rectangular form.
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I forgot to put my e in there.
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I'm going to use this formula e^iθ=cos(θ)+isin(θ).
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That one's really useful, definitely worth remembering.
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This is 8cos(11π/6)+isin(11π/6).
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You could also use x=rcos(θ), y=rsin(θ), you'll end up with the same formula at the end, either way works.
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Let me draw on the unit circle to remind where 11π/6 is.
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11π/6 is just short of 2π, it's right there.
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It's a 30-degree angle south of the x-axis.
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The cosine there is root 3 over 2, it's positive because we're on the right hand side.
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The sine is -1/2.
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What we get there that simplifies down to 4 root 3 minus 4i.
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Now we've done it.
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We've converted each number into polar form.
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We multiplied them in polar form which is very easy, then we converted the polar form back into rectangular form to give us our answer.
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It says we have to check our answer by multiplying them directly in rectangular form.
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Let's do the check here, we'll FOIL the multiplication out.
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I'll do the check over here.
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I'll do the check in blue.
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Foiling it out, my first terms give me -1 times 2 root 3, that's positive root 3.
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My outer terms give me -1-2i, so +2i.
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My inner terms give me -2 times root 3 times root 3, that's 6i.
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Those are my inner terms, I'm doing FOIL here.
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First outer, inner, and my last terms are root 3i minus 2i, that's -2 root 3, i², but i²=-1, this counts as +2 root 3.
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If we simplify that down, we get 2 root 3 plus 2 root 3, 4 root 3, +2i-6i, is -4i.
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That does indeed check with the answer we got by converting into polar form.
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That was kind of a long one.
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Let's recap what we did there.
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We had these two complex numbers.
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We wanted to convert each one into polar form.
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For each one, I found my r, and I used square root of x²+y².
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I found my θ by using arctan(y/x), [intelligible 00:24:55] each one the x's were less than 0, so I had to add on this fudge factor plus π to get me into the right quadrant.
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I found my r, my θ, another r, my other θ.
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I converted there each one into re^iθ form.
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To multiply them together, you multiply the r's but then you add the θ's because they're up in the exponents.
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That's the law of exponents there, so we added the θ's.
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We got a simplified polar form and then we converted back into rectangular form using either the iθ=cos(θ)+isin(θ).
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You could also use x=rcos(θ), y=rsin(θ), you'll get to exactly the same place.
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I know my cosine and sine of 11π/6, that's a common value.
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I get the answer there.
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To check it, I skipped all the polar forms.
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I just multiplied everything out using FOIL, simplified it down and it did indeed check with the answer that I've got using the polar form.
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We'll try some more examples later.
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You should try them on your own first and then we'll work on them together.