This question aims to find the polar coordinates of a point** P** that is equal to **(6, 31°).**

**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 $P (r,\theta)$.

**Expert Answer**

$P (r,\theta)$ 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 $(7.5, 12.99)$.**

*Image/Mathematical drawings are created in Geogebra.*