What is the Cartesian form of (−1, 4π3)?

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What is the Cartesian form of (−1, 4π3)?

Bordadwayjpf · Accepted Answer

Step 1We are given the polar form, so there is a radius and an angle. We want to convert to x,y coordinates.So we can use Euler&#039;s formula (or at least the idea behind it) to convert between Cartesian and polar:x=rcos⁡θy=rsin⁡θFrom that, we just plug in the numbers, remembering our unit circle:cos⁡(4π3)=−12 and sin⁡(4π3)=−32therefore(x,y)=(12,32)You could also notice that a negative radius is the same as adding or subtracting π to the angle, hence(−1,4π3)=(1,π3)which I think is a bit easier to think about.

Anahi Pearson · Accepted Answer

Step 1
The Cartesian coordinates (x,y) of the point $(- 1, \frac{4 π}{3}) \equiv (r, θ)$ are given as follows
$x = r \cos θ$
$= - 1 \cos (\frac{4 π}{3})$
$= - \cos (π + \frac{π}{3})$
$= \cos (\frac{π}{3})$
$= \frac{1}{2}$
$y = r \sin θ$
$= - 1 \sin (\frac{4 π}{3})$
$= - \sin (π + \frac{π}{3})$
$= \sin (\frac{π}{3})$
$= \frac{\sqrt{3}}{2}$
hence, the Cartesian coordinates are $(\frac{1}{2}, \frac{\sqrt{3}}{2})$

What is the Cartesian form of (-1,\ \frac{4\pi}{3})?

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