Prove that if a point 𝐒(x1, y1) has polar coordinates (r, θ), and a point 𝐓(x2, y2) has polar co

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Prove that if a point 𝐒(x1, y1) has polar coordinates (r,...

Prove that if a point $\mathbf{S}\left(x_1, y_1\right)$ has polar coordinates $(r, \theta)$, and a point $\mathbf{T}\left(x_2, y_2\right)$ has polar coordinates $(-r, \theta)$, then $x_1=-x_2$ and $y_1=-y_2$.

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Key Concepts

Polar Coordinates

Polar coordinates represent points in the plane by specifying a distance from the origin and an angle from the positive x-axis. This system is an alternative to the rectangular coordinate system and is especially useful in situations where the geometry of a problem naturally involves distances and directions.

Conversion to Rectangular Coordinates

To translate from polar to rectangular coordinates, the formulas x = r cos(?) and y = r sin(?) are used. These equations serve as the bridge between the polar and Cartesian systems, allowing points defined by distance and angle to be expressed in terms of horizontal and vertical displacements.

Negative Radius

In the polar coordinate system, a negative radius can be interpreted as the point being located in the opposite direction from the angle provided. Specifically, the point with coordinates (?r, ?) effectively represents the reflection of the point (r, ?) through the origin, meaning if the point (r, ?) corresponds to (x, y) in rectangular coordinates, then (?r, ?) corresponds to (?x, ?y).

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