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Ch.12 - Parametric and Polar Curves
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh.12 - Parametric and Polar CurvesProblem 12.2.26
Chapter 12, Problem 12.2.26

25–30. Converting coordinates Express the following polar coordinates in Cartesian coordinates.


(1, 2π/3)

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Recall the formulas to convert polar coordinates \((r, \theta)\) to Cartesian coordinates \((x, y)\): \(x = r \cos(\theta)\) \(y = r \sin(\theta)\)
Identify the given polar coordinates: \(r = 1\) \(\theta = \frac{2\pi}{3}\)
Substitute the values into the formulas: \(x = 1 \times \cos\left(\frac{2\pi}{3}\right)\) \(y = 1 \times \sin\left(\frac{2\pi}{3}\right)\)
Evaluate the trigonometric functions \(\cos\left(\frac{2\pi}{3}\right)\) and \(\sin\left(\frac{2\pi}{3}\right)\) using the unit circle or known values.
Write the Cartesian coordinates as the ordered pair \((x, y)\) using the evaluated values from the previous step.

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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 radius and an angle, denoted as (r, θ), where r is the distance from the origin and θ is the angle measured from the positive x-axis. This system is useful for describing points in circular or rotational contexts.
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Intro to Polar Coordinates

Conversion Formulas from Polar to Cartesian Coordinates

To convert polar coordinates (r, θ) to Cartesian coordinates (x, y), use the formulas x = r cos(θ) and y = r sin(θ). These formulas translate the radius and angle into horizontal and vertical distances on the Cartesian plane.
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Intro to Polar Coordinates

Trigonometric Functions and Angle Measurement

Understanding sine and cosine functions is essential for conversion, as they relate angles to ratios of sides in right triangles. Angles in radians, like 2π/3, must be interpreted correctly to find accurate x and y values.
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Introduction to Trigonometric Functions
Related Practice
Textbook Question

37–52. Curves to parametric equations Find parametric equations for the following curves. Include an interval for the parameter values. Answers are not unique.


The lower half of the circle centered at (−2, 2) with radius 6, oriented in the counterclockwise direction

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Textbook Question

69–72. Tangent lines Find an equation of the line tangent to the following curves at the given point.

y² - x²/64 = 1; (6, -5/4)

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Textbook Question

31–38. Equations of parabolas Find an equation of the following parabolas. Unless otherwise specified, assume the vertex is at the origin.

A parabola symmetric about the y-axis that passes through the point (2, -6)

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Textbook Question

45–60. Areas of regions Find the area of the following regions.


The region inside the limaçon r = 4 - 2 cos θ

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Textbook Question

31–36. Eliminating the parameter Eliminate the parameter to express the following parametric equations as a single equation in x and y.


x=t,y= √(4−t²) a

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Textbook Question

25–28. Horizontal and vertical tangents Find the points at which the following polar curves have horizontal or vertical tangent lines.

r = 4 cos θ

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