Convert the complex number z = 2 ( cos 7 π 4 + i ・ sin 7 π 4 ) z=\sqrt2\left(\cos\frac{7\pi}{4}+i・\sin\frac{7\pi}{4}\right) z = 2 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"> ( cos 4 7 π + i ・ sin 4 7 π ) from polar to rectangular form.

this problem, we are given a complex number written in polar form, and we're asked to convert this from polar to rectangular form. Now to solve these types of problems, the first step to convert to rectangular form should be to take whatever your r value is out in front and distribute it into the parenthesis. So doing this we're going to have that our complex numbers is square root 2 times the cosine of 7pifour++++i square root 2, times the sine of 7pi over 4. So this is what happens when we distribute this r value. Now our next step is going to be to figure out what this cosine and sine of 7 pi over 4 radians is, and you can figure this out by taking a look at the unit circle. 7 pi over 4 radians shows up in the 4th quadrant, And for the cosine, the value is square root 2 over 2. And for the sine, it's negative square root 2 over 2. So rewriting this expression I'll get that Z is equal to the square root of 2 times this value, square root 2 over 2. That's gonna be plus I times square root 2 times this value negative square root 2 over 2. Now at this point all I need to do is evaluate everything and simplify. Now square root of 2 times square root of 2 that's just 2. So we're gonna have 2 over 2. And then from here, well, notice that I have a negative sign in here. I can take that negative sign and I can bring it out to the front and replace it with this plus sign. So we're gonna have 2 over 2 minuteus, and that's going to be I times square root of 2 times square root of 2, which is 2 over 2. Now at this point, notice that the 2's are going to cancel right here and the 2's are going to cancel there. So I'm going to be left with is that our complex number is equal to 1 minus I, and that right there is the complex number in rectangular form and the solution to this problem. So that is how you can solve this type of problem. Hope you found this video helpful. Thanks for watching.

Convert the complex number z = 2 ( cos ⁡ 7 π 4 + i ・ sin ⁡ 7 π 4 ) z=\sqrt2\left(\cos\frac{7\pi}{4}+i・\sin\frac{7\pi}{4}\right) z = 2 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z"> ( cos 4 7 π + i ・ sin 4 7 π ) from polar to rectangular form.

z = 2 − i 2 z=\sqrt2-i\sqrt2 z = 2 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z"> − i 2 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z">

z = 2 − 2 i z=2-2i z = 2 − 2 i

17. Graphing Complex Numbers

Polar Form of Complex Numbers

Precalculus

17. Graphing Complex Numbers

Polar Form of Complex Numbers

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Multiple Choice

Convert the complex number $z=\sqrt2\left(\cos\frac{7\pi}{4}+i・\sin\frac{7\pi}{4}\right)$ from polar to rectangular form.

$z=\sqrt2-i\sqrt2$

$z=1-i$

$z=2-2i$

$z=-i$

Verified step by step guidance

Identify the given complex number in polar form: z = \sqrt{2} \left( \cos \frac{7\pi}{4} + i \sin \frac{7\pi}{4} \right).

Recall that the polar form of a complex number is given by z = r (\cos \theta + i \sin \theta), where r is the magnitude and \theta is the angle.

Convert the angle \frac{7\pi}{4} to its equivalent in the unit circle. Note that \frac{7\pi}{4} is in the fourth quadrant, where cosine is positive and sine is negative.

Calculate the rectangular form using the formulas: x = r \cos \theta and y = r \sin \theta. Here, x and y are the real and imaginary parts, respectively.

Substitute r = \sqrt{2}, \cos \frac{7\pi}{4} = \frac{1}{\sqrt{2}}, and \sin \frac{7\pi}{4} = -\frac{1}{\sqrt{2}} into the formulas to find the rectangular form: z = x + yi.