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Ch.12 - Parametric and Polar Curves
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh.12 - Parametric and Polar CurvesProblema 12.2.2
Capitolo 12, Problema 12.2.2

Write the equations that are used to express a point with polar coordinates (r, θ) in Cartesian coordinates.

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Recall that polar coordinates \((r, \theta)\) represent a point in the plane by its distance \(r\) from the origin and the angle \(\theta\) it makes with the positive x-axis.
To convert from polar to Cartesian coordinates \((x, y)\), we use the relationships based on trigonometry in the right triangle formed by the point, the origin, and the projection on the x-axis.
The x-coordinate is found by projecting the point onto the x-axis, which is given by \(x = r \cos(\theta)\).
Similarly, the y-coordinate is found by projecting the point onto the y-axis, which is given by \(y = r \sin(\theta)\).
Thus, the equations to convert from polar to Cartesian coordinates are: \[ x = r \cos(\theta) \] \[ y = r \sin(\theta) \]

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Polar Coordinates

Polar coordinates represent a point in the plane using a distance from the origin (r) and an angle (θ) measured from the positive x-axis. This system is useful for describing locations in circular or rotational contexts.
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Intro to Polar Coordinates

Cartesian Coordinates

Cartesian coordinates specify a point by its horizontal (x) and vertical (y) distances from the origin along perpendicular axes. This rectangular coordinate system is the standard for graphing and analyzing points in the plane.
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05:32
Intro to Polar Coordinates

Conversion Formulas between Polar and Cartesian Coordinates

To convert from polar to Cartesian coordinates, use the formulas x = r cos(θ) and y = r sin(θ). These equations translate the distance and angle into horizontal and vertical components, enabling the representation of the same point in Cartesian form.
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Intro to Polar Coordinates
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