Complex Numbers in Rectangular, Polar Form - Algebra 2
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Which quadrant contains $z=a+bi$ if $a<0$ and $b>0$?
Which quadrant contains $z=a+bi$ if $a<0$ and $b>0$?
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Quadrant $\text{II}$. Negative real part and positive imaginary part place it in quadrant II.
Quadrant $\text{II}$. Negative real part and positive imaginary part place it in quadrant II.
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What point represents $z=a+bi$ on the complex plane (as an ordered pair)?
What point represents $z=a+bi$ on the complex plane (as an ordered pair)?
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$(a,b)$. Real part gives x-coordinate, imaginary part gives y-coordinate.
$(a,b)$. Real part gives x-coordinate, imaginary part gives y-coordinate.
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What are the conversion formulas from rectangular to polar for $z=a+bi$ (in terms of $r,\theta$)?
What are the conversion formulas from rectangular to polar for $z=a+bi$ (in terms of $r,\theta$)?
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$r=\sqrt{a^2+b^2},\ \tan\theta=\frac{b}{a}$. Use distance formula for $r$ and inverse tangent for $\theta$.
$r=\sqrt{a^2+b^2},\ \tan\theta=\frac{b}{a}$. Use distance formula for $r$ and inverse tangent for $\theta$.
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Convert $z=6\left(\cos\frac{5\pi}{4}+i\sin\frac{5\pi}{4}\right)$ to rectangular form.
Convert $z=6\left(\cos\frac{5\pi}{4}+i\sin\frac{5\pi}{4}\right)$ to rectangular form.
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$-3\sqrt{2}-3\sqrt{2}i$. $\cos(\frac{5\pi}{4})=-\frac{\sqrt{2}}{2}$ and $\sin(\frac{5\pi}{4})=-\frac{\sqrt{2}}{2}$.
$-3\sqrt{2}-3\sqrt{2}i$. $\cos(\frac{5\pi}{4})=-\frac{\sqrt{2}}{2}$ and $\sin(\frac{5\pi}{4})=-\frac{\sqrt{2}}{2}$.
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Find $r$ and a principal $\theta$ for $z=\sqrt{3}-i$ in polar form.
Find $r$ and a principal $\theta$ for $z=\sqrt{3}-i$ in polar form.
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$r=2,\ \theta=-\frac{\pi}{6}$. Quadrant IV with $r=2$ and reference angle $\frac{\pi}{6}$ gives $\theta=-\frac{\pi}{6}$.
$r=2,\ \theta=-\frac{\pi}{6}$. Quadrant IV with $r=2$ and reference angle $\frac{\pi}{6}$ gives $\theta=-\frac{\pi}{6}$.
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What is the principal argument range commonly used for $\operatorname{Arg}(z)$ in Algebra $2$?
What is the principal argument range commonly used for $\operatorname{Arg}(z)$ in Algebra $2$?
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$-\pi<\operatorname{Arg}(z)\le\pi$. Principal value ranges from $-\pi$ to $\pi$ (excluding $-\pi$).
$-\pi<\operatorname{Arg}(z)\le\pi$. Principal value ranges from $-\pi$ to $\pi$ (excluding $-\pi$).
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Convert $z=5\left(\cos\left(-\frac{\pi}{2}\right)+i\sin\left(-\frac{\pi}{2}\right)\right)$ to rectangular form.
Convert $z=5\left(\cos\left(-\frac{\pi}{2}\right)+i\sin\left(-\frac{\pi}{2}\right)\right)$ to rectangular form.
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$0-5i$. $\cos(-\frac{\pi}{2})=0$ and $\sin(-\frac{\pi}{2})=-1$, so $5(0-1i)=-5i$.
$0-5i$. $\cos(-\frac{\pi}{2})=0$ and $\sin(-\frac{\pi}{2})=-1$, so $5(0-1i)=-5i$.
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Identify the ordered pair on the complex plane corresponding to $z=-2+7i$.
Identify the ordered pair on the complex plane corresponding to $z=-2+7i$.
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$(-2,7)$. Complex number maps to point with x-coordinate $-2$, y-coordinate $7$.
$(-2,7)$. Complex number maps to point with x-coordinate $-2$, y-coordinate $7$.
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Identify the ordered pair on the complex plane corresponding to $z=9-3i$.
Identify the ordered pair on the complex plane corresponding to $z=9-3i$.
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$(9,-3)$. Complex number maps to point with x-coordinate $9$, y-coordinate $-3$.
$(9,-3)$. Complex number maps to point with x-coordinate $9$, y-coordinate $-3$.
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Convert $z=2(\cos^0+i\sin^0)$ to rectangular form.
Convert $z=2(\cos^0+i\sin^0)$ to rectangular form.
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$2+0i$. $\cos(0)=1$ and $\sin(0)=0$, so $2(1+0i)=2$.
$2+0i$. $\cos(0)=1$ and $\sin(0)=0$, so $2(1+0i)=2$.
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Convert $z=7(\cos\pi+i\sin\pi)$ to rectangular form.
Convert $z=7(\cos\pi+i\sin\pi)$ to rectangular form.
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$-7+0i$. $\cos(\pi)=-1$ and $\sin(\pi)=0$, so $7(-1+0i)=-7$.
$-7+0i$. $\cos(\pi)=-1$ and $\sin(\pi)=0$, so $7(-1+0i)=-7$.
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Convert $z=3\left(\cos\frac{\pi}{2}+i\sin\frac{\pi}{2}\right)$ to rectangular form.
Convert $z=3\left(\cos\frac{\pi}{2}+i\sin\frac{\pi}{2}\right)$ to rectangular form.
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$0+3i$. $\cos(\frac{\pi}{2})=0$ and $\sin(\frac{\pi}{2})=1$, so $3(0+1i)=3i$.
$0+3i$. $\cos(\frac{\pi}{2})=0$ and $\sin(\frac{\pi}{2})=1$, so $3(0+1i)=3i$.
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Convert $z=5\left(\cos\left(-\frac{\pi}{2}\right)+i\sin\left(-\frac{\pi}{2}\right)\right)$ to rectangular form.
Convert $z=5\left(\cos\left(-\frac{\pi}{2}\right)+i\sin\left(-\frac{\pi}{2}\right)\right)$ to rectangular form.
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$0-5i$. $\cos(-\frac{\pi}{2})=0$ and $\sin(-\frac{\pi}{2})=-1$, so $5(0-1i)=-5i$.
$0-5i$. $\cos(-\frac{\pi}{2})=0$ and $\sin(-\frac{\pi}{2})=-1$, so $5(0-1i)=-5i$.
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Find a principal $\theta$ for $z=-3+3\sqrt{3}i$ (you may leave $r$ unreported).
Find a principal $\theta$ for $z=-3+3\sqrt{3}i$ (you may leave $r$ unreported).
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$\theta=\frac{2\pi}{3}$. Quadrant II with $\tan\theta=-\sqrt{3}$ gives principal angle $\frac{2\pi}{3}$.
$\theta=\frac{2\pi}{3}$. Quadrant II with $\tan\theta=-\sqrt{3}$ gives principal angle $\frac{2\pi}{3}$.
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Convert $z=10\left(\cos\frac{\pi}{6}+i\sin\frac{\pi}{6}\right)$ to rectangular form.
Convert $z=10\left(\cos\frac{\pi}{6}+i\sin\frac{\pi}{6}\right)$ to rectangular form.
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$5\sqrt{3}+5i$. $\cos(\frac{\pi}{6})=\frac{\sqrt{3}}{2}$ and $\sin(\frac{\pi}{6})=\frac{1}{2}$.
$5\sqrt{3}+5i$. $\cos(\frac{\pi}{6})=\frac{\sqrt{3}}{2}$ and $\sin(\frac{\pi}{6})=\frac{1}{2}$.
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What is the rectangular form of a complex number $z$ in terms of $a$ and $b$?
What is the rectangular form of a complex number $z$ in terms of $a$ and $b$?
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$z=a+bi$. Standard form with real part $a$ and imaginary part $b$.
$z=a+bi$. Standard form with real part $a$ and imaginary part $b$.
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What is the meaning of $a$ and $b$ in $z=a+bi$ on the complex plane?
What is the meaning of $a$ and $b$ in $z=a+bi$ on the complex plane?
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$a=\Re(z),\ b=\Im(z)$. $a$ is the real part and $b$ is the imaginary part.
$a=\Re(z),\ b=\Im(z)$. $a$ is the real part and $b$ is the imaginary part.
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What is the polar form of a complex number $z$ using modulus $r$ and angle $\theta$?
What is the polar form of a complex number $z$ using modulus $r$ and angle $\theta$?
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$z=r(\cos\theta+i\sin\theta)$. Uses modulus $r$ and angle $\theta$ with trigonometric functions.
$z=r(\cos\theta+i\sin\theta)$. Uses modulus $r$ and angle $\theta$ with trigonometric functions.
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What is the cis notation for the polar form $r(\cos\theta+i\sin\theta)$?
What is the cis notation for the polar form $r(\cos\theta+i\sin\theta)$?
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$z=r,\text{cis}(\theta)$. Abbreviated notation where cis stands for cosine plus i sine.
$z=r,\text{cis}(\theta)$. Abbreviated notation where cis stands for cosine plus i sine.
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Convert $z=8\left(\cos\frac{\pi}{3}+i\sin\frac{\pi}{3}\right)$ to rectangular form.
Convert $z=8\left(\cos\frac{\pi}{3}+i\sin\frac{\pi}{3}\right)$ to rectangular form.
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$4+4\sqrt{3}i$. $\cos(\frac{\pi}{3})=\frac{1}{2}$ and $\sin(\frac{\pi}{3})=\frac{\sqrt{3}}{2}$.
$4+4\sqrt{3}i$. $\cos(\frac{\pi}{3})=\frac{1}{2}$ and $\sin(\frac{\pi}{3})=\frac{\sqrt{3}}{2}$.
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What is the formula for the modulus of $z=a+bi$?
What is the formula for the modulus of $z=a+bi$?
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$|z|=\sqrt{a^2+b^2}$. Distance formula from origin to point $(a,b)$.
$|z|=\sqrt{a^2+b^2}$. Distance formula from origin to point $(a,b)$.
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Convert $z=6\left(\cos\frac{\pi}{4}+i\sin\frac{\pi}{4}\right)$ to rectangular form.
Convert $z=6\left(\cos\frac{\pi}{4}+i\sin\frac{\pi}{4}\right)$ to rectangular form.
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$3\sqrt{2}+3\sqrt{2}i$. $\cos(\frac{\pi}{4})=\frac{\sqrt{2}}{2}$ and $\sin(\frac{\pi}{4})=\frac{\sqrt{2}}{2}$.
$3\sqrt{2}+3\sqrt{2}i$. $\cos(\frac{\pi}{4})=\frac{\sqrt{2}}{2}$ and $\sin(\frac{\pi}{4})=\frac{\sqrt{2}}{2}$.
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Convert $z=12\left(\cos\frac{2\pi}{3}+i\sin\frac{2\pi}{3}\right)$ to rectangular form.
Convert $z=12\left(\cos\frac{2\pi}{3}+i\sin\frac{2\pi}{3}\right)$ to rectangular form.
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$-6+6\sqrt{3}i$. $\cos(\frac{2\pi}{3})=-\frac{1}{2}$ and $\sin(\frac{2\pi}{3})=\frac{\sqrt{3}}{2}$.
$-6+6\sqrt{3}i$. $\cos(\frac{2\pi}{3})=-\frac{1}{2}$ and $\sin(\frac{2\pi}{3})=\frac{\sqrt{3}}{2}$.
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Convert $z=10\left(\cos\frac{5\pi}{6}+i\sin\frac{5\pi}{6}\right)$ to rectangular form.
Convert $z=10\left(\cos\frac{5\pi}{6}+i\sin\frac{5\pi}{6}\right)$ to rectangular form.
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$-5\sqrt{3}+5i$. $\cos(\frac{5\pi}{6})=-\frac{\sqrt{3}}{2}$ and $\sin(\frac{5\pi}{6})=\frac{1}{2}$.
$-5\sqrt{3}+5i$. $\cos(\frac{5\pi}{6})=-\frac{\sqrt{3}}{2}$ and $\sin(\frac{5\pi}{6})=\frac{1}{2}$.
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Convert $z=8\left(\cos\frac{7\pi}{6}+i\sin\frac{7\pi}{6}\right)$ to rectangular form.
Convert $z=8\left(\cos\frac{7\pi}{6}+i\sin\frac{7\pi}{6}\right)$ to rectangular form.
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$-4\sqrt{3}-4i$. $\cos(\frac{7\pi}{6})=-\frac{\sqrt{3}}{2}$ and $\sin(\frac{7\pi}{6})=-\frac{1}{2}$.
$-4\sqrt{3}-4i$. $\cos(\frac{7\pi}{6})=-\frac{\sqrt{3}}{2}$ and $\sin(\frac{7\pi}{6})=-\frac{1}{2}$.
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Convert $z=6\left(\cos\frac{5\pi}{4}+i\sin\frac{5\pi}{4}\right)$ to rectangular form.
Convert $z=6\left(\cos\frac{5\pi}{4}+i\sin\frac{5\pi}{4}\right)$ to rectangular form.
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$-3\sqrt{2}-3\sqrt{2}i$. $\cos(\frac{5\pi}{4})=-\frac{\sqrt{2}}{2}$ and $\sin(\frac{5\pi}{4})=-\frac{\sqrt{2}}{2}$.
$-3\sqrt{2}-3\sqrt{2}i$. $\cos(\frac{5\pi}{4})=-\frac{\sqrt{2}}{2}$ and $\sin(\frac{5\pi}{4})=-\frac{\sqrt{2}}{2}$.
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Convert $z=12\left(\cos\left(-\frac{\pi}{3}\right)+i\sin\left(-\frac{\pi}{3}\right)\right)$ to rectangular form.
Convert $z=12\left(\cos\left(-\frac{\pi}{3}\right)+i\sin\left(-\frac{\pi}{3}\right)\right)$ to rectangular form.
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$6-6\sqrt{3}i$. $\cos(-\frac{\pi}{3})=\frac{1}{2}$ and $\sin(-\frac{\pi}{3})=-\frac{\sqrt{3}}{2}$.
$6-6\sqrt{3}i$. $\cos(-\frac{\pi}{3})=\frac{1}{2}$ and $\sin(-\frac{\pi}{3})=-\frac{\sqrt{3}}{2}$.
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Find $r$ and a principal $\theta$ for $z=\sqrt{3}+i$ in polar form.
Find $r$ and a principal $\theta$ for $z=\sqrt{3}+i$ in polar form.
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$r=2,\ \theta=\frac{\pi}{6}$. $r=\sqrt{3+1}=2$ and $\tan\theta=\frac{1}{\sqrt{3}}$ gives $\theta=\frac{\pi}{6}$.
$r=2,\ \theta=\frac{\pi}{6}$. $r=\sqrt{3+1}=2$ and $\tan\theta=\frac{1}{\sqrt{3}}$ gives $\theta=\frac{\pi}{6}$.
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Find $r$ and a principal $\theta$ for $z=1+\sqrt{3}i$ in polar form.
Find $r$ and a principal $\theta$ for $z=1+\sqrt{3}i$ in polar form.
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$r=2,\ \theta=\frac{\pi}{3}$. $r=\sqrt{1+3}=2$ and $\tan\theta=\frac{\sqrt{3}}{1}$ gives $\theta=\frac{\pi}{3}$.
$r=2,\ \theta=\frac{\pi}{3}$. $r=\sqrt{1+3}=2$ and $\tan\theta=\frac{\sqrt{3}}{1}$ gives $\theta=\frac{\pi}{3}$.
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Find $r$ and a principal $\theta$ for $z=-\sqrt{3}+i$ in polar form.
Find $r$ and a principal $\theta$ for $z=-\sqrt{3}+i$ in polar form.
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$r=2,\ \theta=\frac{5\pi}{6}$. Quadrant II with $r=2$ and reference angle $\frac{\pi}{6}$ gives $\theta=\frac{5\pi}{6}$.
$r=2,\ \theta=\frac{5\pi}{6}$. Quadrant II with $r=2$ and reference angle $\frac{\pi}{6}$ gives $\theta=\frac{5\pi}{6}$.
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