Textbook Question
25–30. Converting coordinates Express the following polar coordinates in Cartesian coordinates.
(4, 5π)
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16. Parametric Equations & Polar Coordinates
Polar Coordinates
1:50 minutesProblem 12.2.2
Textbook Question
Write the equations that are used to express a point with polar coordinates (r, θ) in Cartesian coordinates.
Verified step by step guidance1
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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Video duration:
1mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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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05:32Intro 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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