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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 $P (r,\theta)$.
Figure 1 – Lines parallel to the y-axis
Figure 2 – Lines parallel to the x-axis
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:
\[ P (x, y) = P (5.143, 3.09) \]
Figure 3 – Polar Coordinate System
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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