Textbook Question
In Exercises 49–58, convert each rectangular equation to a polar equation that expresses r in terms of θ. x = 7
9. Polar Equations
Convert Equations Between Polar and Rectangular Forms
1:50 minutesProblem 5.RE.57
Textbook Question
In Exercises 54–60, convert each polar equation to a rectangular equation. Then use your knowledge of the rectangular equation to graph the polar equation in a polar coordinate system. r = 5 csc θ
Verified step by step guidance1
Recall the relationships between polar and rectangular coordinates: \(x = r \cos \theta\), \(y = r \sin \theta\), and \(r^2 = x^2 + y^2\).
Given the polar equation \(r = 5 \csc \theta\), rewrite \(\csc \theta\) in terms of sine: \(\csc \theta = \frac{1}{\sin \theta}\), so the equation becomes \(r = \frac{5}{\sin \theta}\).
Multiply both sides of the equation by \(\sin \theta\) to get \(r \sin \theta = 5\).
Use the relationship \(y = r \sin \theta\) to substitute and rewrite the equation as \(y = 5\).
Interpret the rectangular equation \(y = 5\): this represents a horizontal line 5 units above the x-axis. In the polar coordinate system, this corresponds to all points where the vertical coordinate is 5.
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Video duration:
1mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Polar and Rectangular Coordinate Systems
Polar coordinates represent points using a radius and an angle (r, θ), while rectangular coordinates use (x, y) positions on a plane. Understanding how these systems relate is essential for converting equations and interpreting graphs in both formats.
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Intro to Polar Coordinates
Conversion Formulas Between Polar and Rectangular Coordinates
Key formulas link polar and rectangular coordinates: x = r cos θ, y = r sin θ, and r² = x² + y². These allow transformation of equations from one system to the other, enabling easier analysis and graphing.
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Trigonometric Functions and Their Reciprocal Identities
The equation involves csc θ, the reciprocal of sin θ (csc θ = 1/sin θ). Recognizing and manipulating these identities helps rewrite the polar equation in terms of x and y, facilitating conversion to rectangular form.
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5:32Fundamental Trigonometric Identities
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