# Find all polar coordinates of point $p = (6, 31°)$.

This question aims to find the polar coordinates of a point $P$ that is equal to $(6,31^{\circ})$.

$P$ is a point on the $xy$ plane. $x$ and $y$ axis are known as the polar axis, while the origin of the $xy$ plane is called the pole. The point $P$ is represented in the form of $P (r,\theta)$.

$P (r,\theta)$ is any point in the $xy$ plane. Distance from pole to point $P$ is $r$ while the angle between the polar axis and $r$ is $\theta$.

To find all the polar coordinates of point P, it needs to be converted into the Cartesian coordinate system, which is also known as the rectangle coordinate system. In a rectangle coordinate system, point $P$ will be written as $P (x, y)$, where $x$ is the distance along the $x-axis$ and $y$ is the distance along the $y-axis$.

Using the trigonometric formulas:

$\cos \theta = \dfrac {x} {r}$

$x = r \cos \theta \quad \quad \quad (i)$

$\sin \theta = \dfrac {y} {r}$

$y = r \sin \theta \quad \quad \quad (ii)$

Putting values of $r = 6$ and $\theta = 31^ {\circ}$ in equation (i), we get:

$x = 6 \cos (31)$

$x = 6 \times 0.8572$

$x = 5.143$

Putting values of $r = 6$ and $\theta = 31^ {\circ}$ in equation (ii), we get:

$y = 6 \sin (31)$

$y = 6 \times 0.515$

$y = 3.09$

Hence,

$P (x, y) = P (5.143, 3.09)$

The polar coordinates of $P(r, \theta)$ are $(5.143, 3.09)$.

## Numerical Solution

The polar coordinates of the Point $P$ at $(6, 31^{\circ})$ are:

$P (x, y) = P (5.143, 3.09)$

## Example

Find all polar coordinates of point $P = (15, 60^ {\circ})$.

Let:

$P (r, \theta) = P (15, 60^ {\circ})$

Using the trigonometric formulas:

$\cos \theta = \dfrac {x} {r}$

$x = r \cos \theta \quad \quad \quad (i)$

$\sin \theta = \dfrac {y} {r}$

$y = r \sin \theta \quad \quad \quad (ii)$

Putting values of $r = 15$ and $\theta = 60^ {\circ}$ in equation (i) and (ii), we get:

$x = 15 \cos (60)$

$x = 15 \times 0.5$

$x = 7.5$

$y = 15 \sin (60)$

$y = 15 \times 0.866$

$y = 12.99$

Hence,

$P (x, y) = P (7.5, 12.99)$

The polar coordinates of $P (r, \theta)$ are $(7.5, 12.99)$.

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