A complex number is a point, so it has a distance and an angle like any other. Written that way, multiplying means multiplying the distances and adding the angles, which is what makes powers easy.
The modulus is the square root of 12 plus 4, which is 4.
The point is up and to the left, so the angle is 150 degrees.
Write it as 4 times cos plus i sin of that angle.
The answer is 4(cos150∘+isin150∘).
Where students go wrong
Taking the argument straight from the calculator. The modulus is always positive, so the angle is what carries the quadrant, and arctangent will not put you in the second or third one on its own.
Also called trigonometric form, De Moivre’s theorem, modulus and argument or cis form.
What you can put on this worksheet
Standard form to trigonometric form
Write in trigonometric form: −3+33i→ 6(cos120∘+isin120∘)
Trigonometric form to standard form
Write in standard form: 6(cos120∘+isin120∘)→ −3+33i
Yes. Every complex numbers in polar form sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.
Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.
Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.
Precalculus is usually taken around 11th or 12th grade, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.
Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.
That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the complex numbers in polar form questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.
Yes. There are two levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.
You pick from 5: standard form to trigonometric form, trigonometric form to standard form, multiplying and dividing, de moivre's theorem and powers in standard form. Tick as many as you want and set how many of each, or let it spread them evenly.
Taking the argument straight from the calculator. The modulus is always positive, so the angle is what carries the quadrant, and arctangent will not put you in the second or third one on its own.