The polar coordinates of a certain point are (r=4.30 cm,...

The polar coordinates of a certain point are $(r=4.30 \mathrm{~cm}$, $\theta=214^{\circ}$ ). (a) Find its Cartesian coordinates $x$ and $y .$ Find the polar coordinates of the points with Cartesian coordinates
(b) $(-x, y)$
(c) $(-2 x,-2 y),$ and
(d) $(3 x,-3 y)$

Key Concepts

Polar Coordinates

Polar coordinates express a point in the plane by specifying its distance from a fixed reference point (the origin) and the angle from a fixed reference direction. This system is particularly useful in scenarios involving circular or rotational symmetry, and it uses a radius (r) and an angle (?) to uniquely determine the position of any point.

Cartesian Coordinates

The Cartesian coordinate system represents points using two perpendicular axes, typically labeled x and y, corresponding to horizontal and vertical positions respectively. This system is widely used in algebra and geometry because it facilitates the description and solution of linear and nonlinear equations through a straightforward grid representation.

Conversion Between Coordinate Systems

Conversion between polar and Cartesian coordinates is essential for solving problems that require switching between these representations. For polar to Cartesian conversion, the formulas x = r cos(?) and y = r sin(?) are applied, while for Cartesian to polar conversion, the formulas r = ?(x² + y²) and ? = arctan(y/x) (with proper quadrant adjustments) are used.

Reflection Transformation

Reflection transformations involve changing the sign of one or more coordinates to reflect a point across an axis. For example, reflecting a point (x, y) to (-x, y) corresponds to reflecting it across the y-axis, which in polar coordinates alters the angle typically by taking its complement or supplement depending on the original position.

Scaling Transformation

Scaling transformations change the distance of a point from the origin by multiplying the coordinates by a constant factor. This operation uniformly expands or contracts the point's position in the plane and affects the polar coordinate system by multiplying the radius by the scaling factor, while the angle remains unchanged if the scaling is applied uniformly in all directions.

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