yego.me
💡 Stop wasting time. Read Youtube instead of watch. Download Chrome Extension

Converting a complex number from polar to rectangular form | Precalculus | Khan Academy


2m read
·Nov 10, 2024

We are told to consider the complex number ( z ), which is equal to the square root of 17 times cosine of 346 degrees plus ( i ) sine of 346 degrees. They ask us to plot ( z ) in the complex plane below. If necessary, round the point coordinates to the nearest integer.

So I encourage you to pause this video and at least think about where we would likely plot this complex number.

All right, now let's work through it together. When you look at it like this, you can see that what's being attempted is a conversion from polar form to rectangular form. If we're thinking about polar form, we can think about the angle of this complex number, which is clearly 346 degrees.

346 degrees would be about... would be about 14 degrees short of a full circle, so it would get us probably something around there. We also see what the magnitude or the modulus of the complex number is right over here: square root of 17.

Square root of 17 is a little bit more than 4 because 4 squared is 16. So if we go in this direction, let's see... that's going to be about 1, 2, 3, 4. We're going to go right about there.

So if I were to just guess where this is going to put us, it's going to put us right around here—right around ( 4 - i ). But let's actually get a calculator out and see if this evaluates to roughly ( 4 - i ).

So for the real part, let's go 346 degrees, and we're going to take the cosine of it, and then we're going to multiply that times the square root of 17. So times 17 square root... a little over four, which is equal to that; actually, yes, the real part does look almost exactly four, especially if we are rounding to the nearest integer; it's a little bit more than four.

Now let's do the imaginary part. So we have 346 degrees, and we're going to take the sine of it, and we're going to multiply that times the square root of 17 times 17 square root... which is equal to... yup, if we were to round to the nearest integer, it's about negative 1.

So we get to this point right over here, which is approximately ( 4 - i ), and we are done.

More Articles

View All
Worked examples: Forms & features of quadratic functions | High School Math | Khan Academy
The function M is given in three equivalent forms, which form most quickly reveals the Y intercept. So let’s just remind ourselves, if I have a function, the graph y is equal to M of x. These are all equivalent forms; they tell us that the function M is g…
The Dark History of ChatGPT
The world was still coming to terms with the powers of the artificial intelligence chatbot called ChatGPT when GPT-4 was released in March of 2023. GPT-4 is miles ahead of GPT-3.5, the engine on which ChatGPT is running. At the time of writing, GPT-4 can …
Eagle Nectar in the Pock | Diggers
There’s something screaming right here. I got to dig this right now! KG and I are in Virginia, hot on the trail of legendary explorer John Smith. We’re trying to make history and be the first to find artifacts from Smith’s 1608 expedition of the Chesapeak…
Lensa makes $1M/Day (& Steals Your Face)
By this point, there’s no doubt about it: artificial intelligence is taking over the mainstream, and people who know how to leverage this technology are getting insanely rich. Applications like Lensa AI and Don AI are literally flipping mobile apps like I…
The Secrets of El Castillo | Buried Truth of the Maya
MEMO: It’s magical just to be here. I’m thinking about how many thousands of stones are overhead, man. So let’s not think a lot about that. KENNY BROAD: My name is Kenny Broad. I’m the mission specialist. NARRATOR: Kenny Broad is a National Geographic e…
Using Fire to Make Tools | The Great Human Race
On this journey, we need to carry grains, milk, water, processing number materials directly on a fire. So, I want to make some clay pots. Prior to the invention of pottery, our ancestors used organic containers such as animal stomachs and baskets to store…