Write the complex number in polar form:

9514 1404 393

  2√30 ∠-120°

Step-by-step explanation:

The modulus is ...

  √((-√30)² +(-3√10)²) = √(30 +90) = √120 = 2√30

The argument is ...

  arctan(-3√10/-√30) = arctan(√3) = -120° . . . . a 3rd-quadrant angle

The polar form of the number can be written as ...

  (2√30)∠-120°

Additional comments

Any of a number of other formats can be used, including ...

  (2√30)cis(-120°)

  (2√30; -120°)

  (2√30; -2π/3)

  2√30·e^(i4π/3)

Of course, the angle -120° (-2π/3 radians) is the same as 240° (4π/3 radians).

__

At least one app I use differentiates between (x, y) and (r; θ) by the use of a semicolon to separate the modulus and argument of polar form coordinates. I find that useful, as a pair of numbers (10.95, 4.19) by itself does not convey the fact that it represents polar coordinates. As you may have guessed, my personal preference is for the notation 10.95∠4.19. (The lack of a ° symbol indicates the angle is in radians.)

312312

step-by-step explanation: