Textbook Question
31–36. Converting coordinates Express the following Cartesian coordinates in polar coordinates in at least two different ways.
(1, √3)
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Ch.12 - Parametric and Polar Curves
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 12, Problem 12.2.28
25–30. Converting coordinates Express the following polar coordinates in Cartesian coordinates.
(2, 7π/4)
Verified step by step guidance1
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:
\(r = 2\)
\(\theta = \frac{7\pi}{4}\)
Substitute the values into the conversion formulas:
\(x = 2 \cos\left(\frac{7\pi}{4}\right)\)
\(y = 2 \sin\left(\frac{7\pi}{4}\right)\)
Evaluate the trigonometric functions \(\cos\left(\frac{7\pi}{4}\right)\) and \(\sin\left(\frac{7\pi}{4}\right)\) using the unit circle or known values for these angles.
Multiply the results by \(r = 2\) to find the Cartesian coordinates \((x, y)\).
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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 (radius r) and an angle θ measured from the positive x-axis. They are expressed as (r, θ), where r ≥ 0 and θ is typically in radians.
Recommended video:
05:32
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 relate the radius and angle to the horizontal and vertical distances from the origin.
Recommended video:
05:32Intro to Polar Coordinates
Trigonometric Values for Common Angles
Understanding the sine and cosine values of common angles, such as 7π/4, is essential for conversion. For example, cos(7π/4) = √2/2 and sin(7π/4) = -√2/2, which help compute exact Cartesian coordinates.
Recommended video:
6:04Introduction 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 upper half of the parabola x=y ², originating at (0, 0)
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Textbook Question
45–60. Areas of regions Find the area of the following regions.
The region inside the lemniscate r² = 6 sin 2θ
Textbook Question
65–68. Eccentricity-directrix approach Find an equation of the following curves, assuming the center is at the origin. Graph the curve, labeling vertices, foci, asymptotes (if they exist), and directrices.
A hyperbola with vertices (0, ±2) and directrices y = ±1
Textbook Question
27–32. Polar curves Graph the following equations.
r = 3 sin 4θ
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Textbook Question
Plot the points with polar coordinates (2, π/6) and (−3, −π/2). Give two alternative sets of coordinate pairs for both points.
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