Ch.12 - Parametric and Polar Curves

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch.12 - Parametric and Polar Curves

Problem 12.2.2

Chapter 12, Problem 12.2.2

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

Verified step by step guidance

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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Key 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.

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.

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

Related Practice

Textbook Question

Golden Gate Bridge Completed in 1937, San Francisco’s Golden Gate Bridge is 2.7 km long and weighs about 890,000 tons. The length of the span between the two central towers is 1280 m; the towers themselves extend 152 m above the roadway. The cables that support the deck of the bridge between the two towers hang in a parabola (see figure). Assuming the origin is midway between the towers on the deck of the bridge, find an equation that describes the cables. How long is a guy wire that hangs vertically from the cables to the roadway 500 m from the center of the bridge?

Textbook Question

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Textbook Question

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Textbook Question

41–44. Intersection points and area Find all the intersection points of the following curves. Find the area of the entire region that lies within both curves

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Textbook Question

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Textbook Question

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