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Write −√3 + i in polar form.

Question

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Write −√3 + i  in polar form.

Accepted Answer

Welcome back. Everyone. In this problem, we want to express the complex number negative three plus the root of three, I in polar form. For our answer choices, A is two multiplied by the root of three, all multiplied by the cosine of a third of pi plus I sine. A third of pi B is 12 all multiplied by the core sign of 5/6 of pi plus I sign 5/6 of pi C is two multiplied by the square root of three all multiplied by the sign of 56 of Pi plus I multiplied by the core sign of +56 of Pi and D is two multiplied by the skirt of three or multiplied by the core sign of 56 of pi plus I multiplied by the sign of +56 of pi. Now, what do we already know? Well, we know that we want to write our number in polar form and here it's in rectangular form. So recall that for a complex number in polar form, that is where Z is written in the form X plus Y I, where X is the real part and Y is the imaginary part. Then in polar form, Z would be equal to R multiplied by the cosine of theta plus I multiplied by the sign of theta where are equals the square root of X squared plus Y squared and theta equals to the arc tangent of Y divided by X. So if I know the values of X and Y, then I'll be able to solve by R and theta and put it in my polar form. No notice that we can tell X and Y from our complex number because X is the real part, which is negative three and Y is the imaginary part which is the square root of three. That means we can know for R and theta, let's start with R. So that tells us R will be the square root of X squared. So that's negative three squared plus the square root of three squared. No negative three squared is nine and the squared of three are squared is three. So R is the square of 12. No, we can simplify that because there's a, there's a, a square number that we can get from that because since 12 equals four multiplied by three, we can rewrite it as the square of four multiplied by the square root of three. And now the square of four equals two. So that tells us R equals two multiplied by the square root of three. Next, let's find theater, no theater. Let me put and the theater would be equal to the arc tangent of Y which is the square of three divided by X, which is negative three. OK. No, what's that? Well, that a tangent, when we were told the angle would actually be 5/6 of pi. So now that we have our value for R and our value for the, that means our number in polar form. So therefore, in polar form me write that properly, our complex number would be two multiplied by the spirit of three all multiplied by the core sign of 56 of pi plus I multiplied by the sign of 5/6 of pi. And when we look back on our answer choices that would have been answer choice. D thanks for watching everyone. I hope this video helped.

_ Write −√3 + i in polar form.

Key Concepts

Polar Coordinates

Modulus of a Complex Number

Argument of a Complex Number

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