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## Algebra (all content)

### Course: Algebra (all content)>Unit 16

Lesson 9: Polar form of complex numbers

# Polar & rectangular forms of complex numbers

Sal rewrites the complex number -3+2i (which is given in rectangular form) in its polar form. Created by Sal Khan.

## Want to join the conversation?

• In the video he just rounds when finding the polar form, but in the problems you have to find exact answers. In the hints, they just jump from from arctan(b/a) to the answer in radians in exact form so I have no idea how to do it.
• We know that...
+------+---------+---------+---------+
|α |π/6 |π/4 |π/3 |
+------+---------+---------+---------+
|sin(α)|1/2 |sqrt(2)/2|sqrt(3)/2|
|cos(α)|sqrt(3)/2|sqrt(2)/2|1/2 |
|tan(α)|sqrt(3)/3|1 |sqrt(3) |
+------+---------+---------+---------+

These numbers and angles should be remembered. One easy way to remember them is that, in the sin row, the number given to sqrt gets incremented with each column, and that on the cos row it decrements with every column. With tan, you can remember that that tan(α) = sin(α)/cos(α).
We can use those values to help us get the exact answer to a problem. For example, let's say that we have 3 - 3i and want to know the angle (α) of this complex number. We know that tan(α) = -3/3 = -1. We can say that α = arctan(tan(α)) but how do we find the exact value of arctan(-1)? We know that an angle of π/4 has a tangent of 1. Therefore, arctan(1) = π/4. Which allows us to conclude that arctan(-1) = -π/4. So, α = arctan(-3/3) = -π/4. Some calculators might do the job for you.
Hope I helped. :)
• Hello
How do I convert the results into exact values?
It´s not clear in the exercises, do I use the trignometric table?

e.g. How do I convert 6*cos(135) to −3√2 ?

It must be simple, but for the life of me I don´t get it ::::::
• Ah, you must familiarize yourself with the unit circle and special triangles, 45-45-90 and 30-60-90.
Because 135 is 90+45, then it would make a 45-45-90 triangle on the II quadrant. From the unit circle we would know cos(45) is √(2)/2 and it is in II quadrant so it's negative. So -√(2)/2*6 = -3√(2).

Sal has videos covered on all these. You can search for it.
• at why does he rewrite the expression in terms of rcostheta+sinthetai? isn't the polar form just (r,theta)?
Why not just write (sqrt13,2.55)? I'm so confused.
• The rectangular representation of a complex number is in the form z = a + bi.

If you were to represent a complex number according to its Cartesian Coordinates, it would be in the form: (a, b); where a, the real part, lies along the x axis and the imaginary part, b, along the y axis.

The Polar Coordinates of a a complex number is in the form (r, θ).

If you want to go from Polar Coordinates to Cartesian Coordinates, that is just:

(r*cos(θ), r*sin(θ)).

Since we saw that the Cartesian Coordinates are (a, b), then:

a = r*cos(θ)
b = r*sin(θ)

And since the rectangular form of a Complex Number is a + bi, just replace the letters:

a + bi = r*cos(θ) + r*sin(θ)i ← The right-hand side is a Complex Number in Polar Form.

Take a look at the difference between a Polar Form and a Polar Coordinate.
• I didnt really get it.. What are radians? And why is he not measuring it in terms of the angles?
• Why does the current video come after this much more complicated video that is earlier in the series: https://www.khanacademy.org/math/precalculus/imaginary_complex_precalc/complex_analysis/v/polar-form-complex-number ? Seems like the current video is a review of that earlier video, but without the way more complicated part regarding exponential form.
• I think I get going from rectangular to polar, but when converting from polar to rectangular, how do you get the exact values of the trig calculations? I always get a decimal, which I don't know how to express precisely.
• if you want to express the rectangular coordinates exactly, you'll have to write in terms of sin and cos unless you have a special angle.
• Is there a video on rectangular to polar form and/or vice versa without it being on the complex plane? For convert point (4, -3) to polar coordinates, or convert 5 angle 170 to rectangular form?
• you would essentially do the same thing without the i being the multiplier(what you are multiplying by) of r sin Θ

alt code for theta: alt 233
• Hello
How do I convert the results into exact values?
It´s not clear in the exercises, do I use the trigonometric table?

e.g. How do I convert 6*cos(135) to −3√2?

PS: I just copied and pasted this but we have a similar question tho and also I want to ask about how to calculate this expression manually (step by step pls)

-√(2)/2*6 = -3√(2)

lastly, what do you mean by "^" when eg. 2^2*5^3=?
• You would use a calculator to do the conversion. Or, it is quite common for teachers to give you a copy of the unit circle and have you memorize the cosine and sine values of common angles. 135 degrees is one of those common angles. You can find diagrams of the unit circle if you do a websearch and use that as a reference.

The carat symbol ^ is used to denote an exponent since exponents are not easily typed.
2^2 * 5^3 means 2-square times 5-cubed.

Hope this helps.
(1 vote)
• Hello, I was wondering how I'm supposed to know the exact value of certain things, for example square roots, because my calculator only gives answers in decimals
• In the Practice problems Polar and Rectangular forms of complex numbers, Express z1=3-3i in polar form, how do you get the answer arctan of (-3/3) = pi over 4? Is there a calculator that will display the answer as this? My calculator displays arctan (-3/3) as -0.785