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Polar coordinates represent a point in a plane by its distance from a reference point and an angle from a reference direction. This system uses two values: the radius (\r\textunderscore) and the angle (\theta\textunderscore). The radius is the straight-line distance from the origin to the point, while the angle is measured from the positive x-axis to the line segment connecting the origin to the point. Polar coordinates are usually written as \(\r, \theta\). Some key features:

• The radius \(r\) can be any non-negative number.
• The angle \(\theta\) can be measured in degrees (°) or radians.
• Common angles you might encounter include \180^{\circ}\, \90^{\circ}\, and \270^{\circ}\.

For example, in the problem given, \ (4, 180^{\circ})\ represents a point 4 units away from the origin, and the angle of \180^{\circ}\ means it is directly on the negative x-axis.

Rectangular coordinates, also known as Cartesian coordinates, describe a point using a pair of values (\(x, y\)). This system is based on orthogonal axes: the horizontal x-axis and the vertical y-axis. Each point is determined by how far it is from these two axes. Some important considerations:

• \(x\) measures the horizontal distance from the y-axis.
• \(y\) measures the vertical distance from the x-axis.
• The coordinates are written in the form (\(x, y\)).

To convert a point from polar to rectangular coordinates, we use trigonometric functions to find these distances. This involves finding how much the point moves horizontally (\(x\)) and vertically (\(y\)) relative to the origin. As an example, for the point \( (4, 180^{\circ}) \), the rectangular coordinates are found using the equations:
\(x = 4 \cos(180^{\circ}) = -4\)
\(y = 4 \sin(180^{\circ}) = 0\).
Therefore, the rectangular coordinates are \((-4, 0)\).

Trigonometric functions help us relate the angles of a triangle to its sides, which is crucial when converting polar coordinates to rectangular coordinates. The main trigonometric functions used are cosine and sine, denoted as \(cos\) and \(sin\) respectively.

• \(cos(\theta)\) gives the ratio of the adjacent side to the hypotenuse in a right triangle.
• \(sin(\theta)\) gives the ratio of the opposite side to the hypotenuse.

In polar to rectangular conversion for a point \((r,\theta)\):

• \(x = r \cos(\theta)\) calculates the horizontal distance.
• \(y = r \sin(\theta).\) calculates the vertical distance.

For the point \( (4, 180^{\circ}) \),

• \(cos(180^{\circ}) = -1\) since 180 degrees is a straight line on the negative x-axis.
• \(sin(180^{\circ}) = 0\) because there is no vertical component.

Putting these in the formulas:
\(x = 4 \cdot (-1) = -4\) and \(y = 4 \cdot 0 = 0\).
So, trigonometric functions help us find that the rectangular coordinates are (\(-4, 0)\).