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Polar coordinates are a method for describing the location of a point on a plane. A point in polar coordinates is given as \((r, \theta)\), where \(r\) and \(\theta\) have distinct roles.- **Radial Distance \(r\):** This is the distance from the origin to the point. It is always a non-negative number, denoting how far the point is from the center of the plane.- **Angle \(\theta\):** This angle is measured in degrees (or radians) and determines the direction from the positive x-axis at which the point lies.When reading polar coordinates like \((5, 270^{\circ})\), the number 5 is how far the point is from the center, and the \(270^{\circ}\) angle tells you the direction you'll be moving from the x-axis. In this context, \(270^{\circ}\) signifies three quarters of the way around the circle, pointing directly downwards along the negative y-axis. Understanding these two components helps in converting these polar points to their rectangular counterparts.

Rectangular coordinates, or Cartesian coordinates, denote a point using an \(x\) and \(y\) value, which relate to distances along the horizontal (x-axis) and vertical (y-axis) directions, respectively.- **x-coordinate \(x\):** This specifies the horizontal distance from the origin. It can be positive or negative, showing the direction along the x-axis. Positive values move right, while negative ones move left.- **y-coordinate \(y\):** This specifies the vertical distance from the origin. Positive values move upwards, while negative ones move downwards.For instance, after converting the polar coordinates \((5, 270^{\circ})\) into rectangular coordinates, you get \((x, y) = (0, -5)\). This tells you that the point lies directly on the negative y-axis, 5 units downward from the origin. Rectangular coordinates give a clear way to map a point's precise location in terms of horizontal and vertical placement on a plane.

Converting between polar and rectangular coordinates involves using simple trigonometric relationships. The key formulas used are:- **x-coordinate formula:** \(x = r \cos(\theta)\) - **y-coordinate formula:** \(y = r \sin(\theta)\)These formulas rely on the concepts of cosine and sine from trigonometry:- **Cosine \(\cos(\theta)\):** Relates the adjacent side to the hypotenuse in a right triangle, here giving the horizontal component.- **Sine \(\sin(\theta)\):** Relates the opposite side to the hypotenuse, providing the vertical component.For \((5, 270^{\circ})\), using \(x = 5 \cos(270^{\circ})\), since \(\cos(270^{\circ}) = 0\), we derive that \(x = 0\). Similarly, for \(y = 5 \sin(270^{\circ})\), since \(\sin(270^{\circ}) = -1\), we find \(y = -5\). These formulas effectively translate the radial distance and angular direction into definitive horizontal and vertical components, making it easy to switch between the two systems of coordinates.