Complex Numbers in Rectangular, Polar Form - Algebra 2

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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