Algebra 2 Flashcards: Complex Numbers In Rectangular Polar Form

What this deck covers

This deck focuses on Complex Numbers In Rectangular Polar Form, giving you a quick way to review the definitions, rules, and examples that matter most for Algebra 2.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: Identify the correct principal angle for $z=-1+i$ in polar form.

Answer: $\frac{3\pi}{4}$. Quadrant II with equal components gives angle $\frac{3\pi}{4}$.

Flashcard 2: What is the polar form of the imaginary number $4i$ using a principal argument?

Answer: $4\left(\cos\frac{\pi}{2}+i\sin\frac{\pi}{2}\right)$. Pure imaginary number $4i$ has angle $\frac{\pi}{2}$.

Flashcard 3: Find the modulus $|z|$ for $z=3+4i$.

Answer: $5$. $\sqrt{3^2+4^2}=\sqrt{9+16}=\sqrt{25}=5$

Flashcard 4: Which quadrant contains $z=a+bi$ if $a<0$ and $b>0$?

Answer: Quadrant $\text{II}$. Negative real part and positive imaginary part place it in quadrant II.

Flashcard 5: Convert $z=10\left(\cos\frac{\pi}{6}+i\sin\frac{\pi}{6}\right)$ to rectangular form.

Answer: $5\sqrt{3}+5i$. $\cos(\frac{\pi}{6})=\frac{\sqrt{3}}{2}$ and $\sin(\frac{\pi}{6})=\frac{1}{2}$.

Flashcard 6: What is $\operatorname{Arg}(z)$ if $z$ is a positive imaginary number $bi$ with $b>0$?

Answer: $\frac{\pi}{2}$. Positive imaginary numbers lie on the positive y-axis.

Flashcard 7: Convert $z=6\left(\cos\frac{\pi}{4}+i\sin\frac{\pi}{4}\right)$ to rectangular form.

Answer: $3\sqrt{2}+3\sqrt{2}i$. $\cos(\frac{\pi}{4})=\frac{\sqrt{2}}{2}$ and $\sin(\frac{\pi}{4})=\frac{\sqrt{2}}{2}$.

Flashcard 8: What are $\cos\theta$ and $\sin\theta$ in terms of $a,b,r$ for $z=a+bi$ with $r=|z|$?

Answer: $\cos\theta=\frac{a}{r},\ \sin\theta=\frac{b}{r}$. Unit circle relationships connect rectangular and polar coordinates.

Flashcard 9: Which quadrant contains $z=a+bi$ if $a<0$ and $b<0$?

Answer: Quadrant $\text{III}$. Both real and imaginary parts negative place it in quadrant III.

Flashcard 10: Find a principal $\theta$ for $z=3-3\sqrt{3}i$ (you may leave $r$ unreported).

Answer: $\theta=-\frac{\pi}{3}$. Quadrant IV with $\tan\theta=-\sqrt{3}$ gives principal angle $-\frac{\pi}{3}$.

Flashcard 11: Find $r$ and a principal $\theta$ for $z=\sqrt{3}-i$ in polar form.

Answer: $r=2,\ \theta=-\frac{\pi}{6}$. Quadrant IV with $r=2$ and reference angle $\frac{\pi}{6}$ gives $\theta=-\frac{\pi}{6}$.

Flashcard 12: What is the polar form of a complex number $z$ using modulus $r$ and angle $\theta$?

Answer: $z=r(\cos\theta+i\sin\theta)$. Uses modulus $r$ and angle $\theta$ with trigonometric functions.

Flashcard 13: What is the polar form of the real number $-5$ using a principal argument?

Answer: $5(\cos\pi+i\sin\pi)$. Negative real number has angle $\pi$ and modulus 5.

Flashcard 14: What is $\operatorname{Arg}(z)$ if $z$ is a negative imaginary number $bi$ with $b<0$?

Answer: $-\frac{\pi}{2}$. Negative imaginary numbers lie on the negative y-axis.

Flashcard 15: Convert $z=10\left(\cos\frac{5\pi}{6}+i\sin\frac{5\pi}{6}\right)$ to rectangular form.

Answer: $-5\sqrt{3}+5i$. $\cos(\frac{5\pi}{6})=-\frac{\sqrt{3}}{2}$ and $\sin(\frac{5\pi}{6})=\frac{1}{2}$.

Flashcard 16: Find the modulus $|z|$ for $z=5-12i$.

Answer: $13$. $\sqrt{5^2+(-12)^2}=\sqrt{25+144}=\sqrt{169}=13$

Flashcard 17: What is the relationship between all arguments of a nonzero complex number $z$ and its principal argument $\operatorname{Arg}(z)$?

Answer: $\arg(z)=\operatorname{Arg}(z)+2\pi k$. All arguments differ by multiples of $2\pi$.

Flashcard 18: Find $r$ and a principal $\theta$ for $z=1+\sqrt{3}i$ in polar form.

Answer: $r=2,\ \theta=\frac{\pi}{3}$. $r=\sqrt{1+3}=2$ and $\tan\theta=\frac{\sqrt{3}}{1}$ gives $\theta=\frac{\pi}{3}$.

Flashcard 19: What is the meaning of $a$ and $b$ in $z=a+bi$ on the complex plane?

Answer: $a=\Re(z),\ b=\Im(z)$. $a$ is the real part and $b$ is the imaginary part.

Flashcard 20: What is the geometric meaning of $|z|$ for $z=a+bi$ on the complex plane?

Answer: Distance from $(0,0)$ to $(a,b)$. Represents the distance from the origin to the point.

Flashcard 21: Find $r$ and a principal $\theta$ for $z=-1+\sqrt{3}i$ in polar form.

Answer: $r=2,\ \theta=\frac{2\pi}{3}$. Quadrant II with $r=2$ and reference angle $\frac{\pi}{3}$ gives $\theta=\frac{2\pi}{3}$.

Flashcard 22: What is the principal argument range commonly used for $\operatorname{Arg}(z)$ in Algebra $2$?

Answer: $-\pi<\operatorname{Arg}(z)\le\pi$. Principal value ranges from $-\pi$ to $\pi$ (excluding $-\pi$).

Flashcard 23: Which quadrant contains $z=a+bi$ if $a>0$ and $b<0$?

Answer: Quadrant $\text{IV}$. Positive real part and negative imaginary part place it in quadrant IV.

Flashcard 24: Find $r$ and a principal $\theta$ for $z=-\sqrt{3}-i$ in polar form.

Answer: $r=2,\ \theta=-\frac{5\pi}{6}$. Quadrant III with $r=2$ and reference angle $\frac{\pi}{6}$ gives $\theta=-\frac{5\pi}{6}$.

Flashcard 25: Find the modulus $|z|$ for $z=-8-6i$.

Answer: $10$. $\sqrt{(-8)^2+(-6)^2}=\sqrt{64+36}=\sqrt{100}=10$

Flashcard 26: What is the polar form of the imaginary number $-4i$ using a principal argument?

Answer: $4\left(\cos\left(-\frac{\pi}{2}\right)+i\sin\left(-\frac{\pi}{2}\right)\right)$. Pure imaginary number $-4i$ has angle $-\frac{\pi}{2}$.

Flashcard 27: Convert $z=5\left(\cos\left(-\frac{\pi}{2}\right)+i\sin\left(-\frac{\pi}{2}\right)\right)$ to rectangular form.

Answer: $0-5i$. $\cos(-\frac{\pi}{2})=0$ and $\sin(-\frac{\pi}{2})=-1$, so $5(0-1i)=-5i$.

Flashcard 28: What is the formula for the modulus of $z=a+bi$?

Answer: $|z|=\sqrt{a^2+b^2}$. Distance formula from origin to point $(a,b)$.

Flashcard 29: What are the conversion formulas from polar to rectangular for $z=r(\cos\theta+i\sin\theta)$?

Answer: $a=r\cos\theta,\ b=r\sin\theta$. Trigonometric relationships for converting polar to rectangular.

Flashcard 30: What is the rectangular form of a complex number $z$ in terms of $a$ and $b$?

Answer: $z=a+bi$. Standard form with real part $a$ and imaginary part $b$.

Flashcard 31: Find $r$ and a principal $\theta$ for $z=1-\sqrt{3}i$ in polar form.

Answer: $r=2,\ \theta=-\frac{\pi}{3}$. Quadrant IV with $r=2$ and reference angle $\frac{\pi}{3}$ gives $\theta=-\frac{\pi}{3}$.

Flashcard 32: What is $\operatorname{Arg}(z)$ if $z$ is a negative real number?

Answer: $\pi$. Negative real numbers lie on the negative x-axis.

Flashcard 33: Find the modulus $|z|$ for $z=-6+8i$.

Answer: $10$. $\sqrt{(-6)^2+8^2}=\sqrt{36+64}=\sqrt{100}=10$

Flashcard 34: Find $r$ and a principal $\theta$ for $z=\sqrt{3}+i$ in polar form.

Answer: $r=2,\ \theta=\frac{\pi}{6}$. $r=\sqrt{3+1}=2$ and $\tan\theta=\frac{1}{\sqrt{3}}$ gives $\theta=\frac{\pi}{6}$.

Flashcard 35: Find a principal $\theta$ for $z=-3+3\sqrt{3}i$ (you may leave $r$ unreported).

Answer: $\theta=\frac{2\pi}{3}$. Quadrant II with $\tan\theta=-\sqrt{3}$ gives principal angle $\frac{2\pi}{3}$.

Flashcard 36: Find $r$ and a principal $\theta$ for $z=-\sqrt{3}+i$ in polar form.

Answer: $r=2,\ \theta=\frac{5\pi}{6}$. Quadrant II with $r=2$ and reference angle $\frac{\pi}{6}$ gives $\theta=\frac{5\pi}{6}$.

Flashcard 37: What is the cis notation for the polar form $r(\cos\theta+i\sin\theta)$?

Answer: $z=r\,\text{cis}(\theta)$. Abbreviated notation where cis stands for cosine plus i sine.

Flashcard 38: State the identity that explains why $r(\cos\theta+i\sin\theta)$ equals $a+bi$ when $a=r\cos\theta$ and $b=r\sin\theta$.

Answer: $r(\cos\theta+i\sin\theta)=r\cos\theta+i(r\sin\theta)$. Distributive property shows both forms represent the same complex number.

Flashcard 39: Identify the correct principal angle for $z=-1-i$ in polar form.

Answer: $-\frac{3\pi}{4}$. Quadrant III with equal components gives angle $-\frac{3\pi}{4}$.

Flashcard 40: What is the polar form of the real number $5$ using a principal argument?

Answer: $5(\cos^0+i\sin^0)$. Real number 5 has angle 0 and modulus 5.

Flashcard 41: Convert $z=8\left(\cos\frac{7\pi}{6}+i\sin\frac{7\pi}{6}\right)$ to rectangular form.

Answer: $-4\sqrt{3}-4i$. $\cos(\frac{7\pi}{6})=-\frac{\sqrt{3}}{2}$ and $\sin(\frac{7\pi}{6})=-\frac{1}{2}$.

Flashcard 42: Convert $z=3\left(\cos\frac{\pi}{2}+i\sin\frac{\pi}{2}\right)$ to rectangular form.

Answer: $0+3i$. $\cos(\frac{\pi}{2})=0$ and $\sin(\frac{\pi}{2})=1$, so $3(0+1i)=3i$.

Flashcard 43: Convert $z=8\left(\cos\frac{\pi}{3}+i\sin\frac{\pi}{3}\right)$ to rectangular form.

Answer: $4+4\sqrt{3}i$. $\cos(\frac{\pi}{3})=\frac{1}{2}$ and $\sin(\frac{\pi}{3})=\frac{\sqrt{3}}{2}$.

Flashcard 44: What point represents $z=a+bi$ on the complex plane (as an ordered pair)?

Answer: $(a,b)$. Real part gives x-coordinate, imaginary part gives y-coordinate.

Flashcard 45: What is $\operatorname{Arg}(z)$ if $z$ is a positive real number (and $z\ne^0$)?

Answer: $0$. Positive real numbers lie on the positive x-axis.

Flashcard 46: What are the conversion formulas from rectangular to polar for $z=a+bi$ (in terms of $r,\theta$)?

Answer: $r=\sqrt{a^2+b^2},\ \tan\theta=\frac{b}{a}$. Use distance formula for $r$ and inverse tangent for $\theta$.

Flashcard 47: Find $r$ and a principal $\theta$ for $z=-1-\sqrt{3}i$ in polar form.

Answer: $r=2,\ \theta=-\frac{2\pi}{3}$. Quadrant III with $r=2$ and reference angle $\frac{\pi}{3}$ gives $\theta=-\frac{2\pi}{3}$.

Flashcard 48: Identify the ordered pair on the complex plane corresponding to $z=9-3i$.

Answer: $(9,-3)$. Complex number maps to point with x-coordinate $9$, y-coordinate $-3$.

Flashcard 49: Convert $z=12\left(\cos\frac{2\pi}{3}+i\sin\frac{2\pi}{3}\right)$ to rectangular form.

Answer: $-6+6\sqrt{3}i$. $\cos(\frac{2\pi}{3})=-\frac{1}{2}$ and $\sin(\frac{2\pi}{3})=\frac{\sqrt{3}}{2}$.

Flashcard 50: Convert $z=12\left(\cos\left(-\frac{\pi}{3}\right)+i\sin\left(-\frac{\pi}{3}\right)\right)$ to rectangular form.

Answer: $6-6\sqrt{3}i$. $\cos(-\frac{\pi}{3})=\frac{1}{2}$ and $\sin(-\frac{\pi}{3})=-\frac{\sqrt{3}}{2}$.

Flashcard 51: Convert $z=6\left(\cos\frac{5\pi}{4}+i\sin\frac{5\pi}{4}\right)$ to rectangular form.

Answer: $-3\sqrt{2}-3\sqrt{2}i$. $\cos(\frac{5\pi}{4})=-\frac{\sqrt{2}}{2}$ and $\sin(\frac{5\pi}{4})=-\frac{\sqrt{2}}{2}$.

Flashcard 52: Identify the ordered pair on the complex plane corresponding to $z=-2+7i$.

Answer: $(-2,7)$. Complex number maps to point with x-coordinate $-2$, y-coordinate $7$.

Flashcard 53: Convert $z=2(\cos^0+i\sin^0)$ to rectangular form.

Answer: $2+0i$. $\cos(0)=1$ and $\sin(0)=0$, so $2(1+0i)=2$.

Flashcard 54: Convert $z=7(\cos\pi+i\sin\pi)$ to rectangular form.

Answer: $-7+0i$. $\cos(\pi)=-1$ and $\sin(\pi)=0$, so $7(-1+0i)=-7$.