Krystal Villanueva

2022-02-02

What is the distance between the following polar coordinates?:

### Answer & Explanation

stefjumnmt

Step 1
The distance formula for polar coordinates is
$d=\sqrt{{r}_{1}^{2}{r}_{2}^{2}-2{r}_{1}{r}_{2}\mathrm{cos}\left({\theta }_{1}-{\theta }_{2}\right)}$
Where d is the distance between the two points, ${r}_{1}$, and ${\theta }_{1}$ are the polar coordinates of one point and ${r}_{2}$ and ${\theta }_{2}$ are the polar coordinates of another point.
Let represent and represent
$⇒d=\sqrt{{4}^{2}+{5}^{2}-2×4×5\mathrm{cos}\left(\pi -\pi \right)}$
$⇒d=\sqrt{16+25-40\mathrm{cos}\left(0\right)}$
$⇒d=\sqrt{41-40×1}=\sqrt{41-40}=\sqrt{1}=1$
$⇒d=1$
Hence the distance between the given points is 1.

Iacopelli5co

Step 1

Using common insight rather than applying the Pythagorean Theorem and cos conversions:
The distance between any two polar coordinates with the same angle is the difference in their radii.

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