Table of contents

0. Review of College Algebra4h 43m
1. Measuring Angles40m
2. Trigonometric Functions on Right Triangles2h 5m
3. Unit Circle1h 19m
4. Graphing Trigonometric Functions1h 19m
5. Inverse Trigonometric Functions and Basic Trigonometric Equations1h 41m
6. Trigonometric Identities and More Equations2h 34m
7. Non-Right Triangles1h 38m
8. Vectors2h 25m
9. Polar Equations2h 5m
10. Parametric Equations1h 6m
11. Graphing Complex Numbers1h 7m

9. Polar Equations

Polar Coordinate System

Trigonometry

9. Polar Equations

Polar Coordinate System

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2:19 minutes

Problem 51

Textbook Question

In Exercises 49–58, convert each rectangular equation to a polar equation that expresses r in terms of θ.x = 7

Verified step by step guidance

Start with the given rectangular equation: \( x = 7 \).

Recall the relationship between rectangular and polar coordinates: \( x = r \cos \theta \) and \( y = r \sin \theta \).

Substitute the polar coordinate expression for \( x \) into the equation: \( r \cos \theta = 7 \).

Solve for \( r \) in terms of \( \theta \) by dividing both sides by \( \cos \theta \): \( r = \frac{7}{\cos \theta} \).

Recognize that \( \frac{1}{\cos \theta} \) is equivalent to \( \sec \theta \), so the polar equation becomes \( r = 7 \sec \theta \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, represent points in a two-dimensional space using an ordered pair (x, y). In this system, 'x' denotes the horizontal distance from the origin, while 'y' indicates the vertical distance. Understanding how these coordinates relate to polar coordinates is essential for converting equations between the two systems.

Polar Coordinates

Polar coordinates describe a point in a plane using a distance from a reference point (the origin) and an angle from a reference direction (usually the positive x-axis). A point is represented as (r, θ), where 'r' is the radial distance and 'θ' is the angle. Converting from rectangular to polar coordinates involves using the relationships r = √(x² + y²) and θ = arctan(y/x).

Conversion Formulas

To convert rectangular equations to polar form, specific formulas are used. The key relationships are x = r cos(θ) and y = r sin(θ). For the given equation x = 7, substituting the polar equivalent gives r cos(θ) = 7, which can be rearranged to express r in terms of θ, facilitating the conversion process.