Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 10: Problem 91
Convert the polar equation to rectangular form. $$r=4 \sin \theta$$
Short Answer
Expert verified
\The rectangular form of the polar equation \(r = 4 \sin \theta\) is \(y = 4\sin^2\theta - x^2\)
Step by step solution
01
Identify relationship between variables
The relation between polar coordinates (r, 胃) and rectangular coordinates (x, y) can be expressed as follows: \(x= r \cos \theta\) and \(y= r \sin \theta\). In this particular case, the given polar equation is \(r = 4 \sin \theta\). The getKey here is to isolate y, the variable we're trying to express in rectangular coordinates.
02
Substitute \(r\) in y = r sin(胃)
Substitute \(r\) (which is \(4 \sin \theta\)) into \(y= r \sin \theta\) to get rectangular form. It is known that \(y= r \sin \theta\), therefore by substituting you get \(y = 4 \sin^2 \theta\).
03
Use trigonometric identity
To reduce the complexity of the function, we substitute \(\sin^2 \theta\) with the equivalent expression using the Pythagorean trigonometric identity, which is \(1 - cos^2 \theta\). So, \(y = 4(1 - \cos^2 \theta)\)
04
Substitute \(cos \theta\) in rectangular form
Now substitute \(\cos \theta\) with its rectangular equation \(cos \theta = \frac{x}{r}\). By doing so, you get \(y = 4(1 - \left(\frac{x}{r}\right)^2)\), which simplifies to \(y = 4(1 - \left(\frac{x}{4 \sin \theta}\right)^2)\)
05
Simplify the equation further
This relationship can now be simplified further as \(y = 4(\sin^2\theta - (\frac{x^2\sin^2\theta}{16} )\)
06
Simplify to final form
Finally, simplify the equation further to \(y = 4\sin^2\theta - x^2\). This is the equation in rectangular form
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Polar Coordinates
Polar coordinates, a system used to define positions on a plane, center around a point called the origin. Each point is identified by a distance from the origin, known as radius \(r\), and an angle \(\theta\), measured from the positive x-axis. This method is particularly efficient for problems involving circular or spiral forms, where angles play a critical role.
Unlike Cartesian coordinates鈥攚hich specify points by horizontal and vertical distances from the origin鈥攑olar coordinates use:
Unlike Cartesian coordinates鈥攚hich specify points by horizontal and vertical distances from the origin鈥攑olar coordinates use:
- \(r\): Distance from the origin.
- \(\theta\): Angle with respect to the positive x-axis.
Rectangular Coordinates
Rectangular coordinates, also known as Cartesian coordinates, utilize a grid defined by perpendicular x and y axes to locate points. These coordinates make calculations straightforward for linear movements but can be cumbersome for angles or circular paths.
Key characteristics include:
Key characteristics include:
- \(x\): The horizontal distance from the origin.
- \(y\): The vertical distance from the origin.
Trigonometric Identity
Trigonometric identities are equations involving trigonometric functions that are true for all angles \(\theta\). They play a crucial role in converting between different coordinate systems by simplifying equations.
Some commonly used trigonometric identities include:
Some commonly used trigonometric identities include:
- \(\sin^2 \theta + \cos^2 \theta = 1\) (Pythagorean Identity)
- \(\sin \theta = \frac{opposite}{hypotenuse}\)
- \(\cos \theta = \frac{adjacent}{hypotenuse}\)
Rectangular Form Equations
Transforming a polar equation into a rectangular form involves re-expressing terms using equivalent rectangular coordinates. This process can simplify solving and understanding the problem.
To convert \(r = 4 \sin \theta\) into a rectangular form, we:
To convert \(r = 4 \sin \theta\) into a rectangular form, we:
- Recognize that \(y = r \sin \theta = 4 \sin^2 \theta\).
- Apply the Pythagorean identity \(\sin^2 \theta = 1 - \cos^2 \theta\), to express it as \(4(1 - \left(\frac{x}{r}\right)^2 )\).
