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Polar coordinates consist of two components: a radial distance and an angular measurement. These coordinates are expressed as \((r, \theta)\). Here, \(r\) represents how far the point is from the origin (the center point of a plane), while \(\theta\) is the angle formed with the positive x-axis.
Polar coordinates are particularly useful in contexts where directions and magnitudes are naturally expressed in radial forms, such as in navigation or when describing circular motion.
For example, in the given problem, the polar coordinates \((-4, \frac{3\pi}{2})\) indicate a point with a radial distance of \(-4\) and an angle of \(\frac{3\pi}{2}\) radians. The negative sign of \(r\) signifies that the point is in the opposite direction along the line from the origin based on the given angle.

Rectangular coordinates, also known as Cartesian coordinates, are a way of representing points in a 2-dimensional plane. They are given in the form \((x, y)\), where \(x\) represents the horizontal distance from the origin along the x-axis, and \(y\) represents the vertical distance along the y-axis.
These coordinates are highly intuitive in a grid-like system, making them ideal for tasks involving alignment and scaling.
In the solution above, the polar coordinates \((-4, \frac{3\pi}{2})\) were converted to rectangular coordinates as \((0, 4)\). Here, \(x = 0\) places the point directly on the y-axis, and \(y = 4\) indicates that the point is four units above the origin on the y-axis.

To convert polar coordinates to rectangular coordinates, we use specific formulas. These formulas are based on basic trigonometric relationships. For a given point \((r, \theta)\):
\[ x = r \cos(\theta) \]
\[ y = r \sin(\theta) \]
These formulas translate the radial distance \(r\) and angle \(\theta\) into horizontal and vertical distances.
In the exercise, the formulas were used to compute:

• \(x = -4 \cos\left(\frac{3\pi}{2}\right) = 0\)
• \(y = -4 \sin\left(\frac{3\pi}{2}\right) = 4\)

Thus, the conversion results in the rectangular coordinate \((0, 4)\).

Trigonometric functions, namely sine and cosine, are essential in the conversion of polar coordinates to rectangular coordinates. They relate the angles to ratios of the sides in a right triangle.
Understanding these functions is crucial since they determine the \(x\) and \(y\) components based on the angle \(\theta\).
The cosine function \(\cos(\theta)\) gives the ratio of the adjacent side to the hypotenuse, thereby calculating the horizontal component \(x\). Similarly, the sine function \(\sin(\theta)\) gives the ratio of the opposite side to the hypotenuse, thus determining the vertical component \(y\).
In the conversion problem, \(\cos\left(\frac{3\pi}{2}\right) = 0\) and \(\sin\left(\frac{3\pi}{2}\right) = -1\) provide crucial values that transform the polar coordinates to \((0, 4)\).