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Chapter 6: Problem 80

For each rectangular equation, write an equivalent polar equation. $$ y^{2}=2 x $$

Short Answer

Expert verified
The equivalent polar equation is \[ r = \frac{2 \, \cos(\theta)}{\sin^2(\theta)} \]

Step by step solution

01

Recall the conversion formulas

To convert from rectangular to polar coordinates, use the following formulas: \[ x = r \, \cos(\theta) \] \[ y = r \, \sin(\theta) \]
02

Substitute rectangular coordinates with polar coordinates

Given the rectangular equation \[ y^{2}=2x \]Substitute the conversion formulas: \[ (r \, \sin(\theta))^2 = 2 (r \, \cos(\theta)) \]
03

Simplify the polar equation

Simplify the equation by squaring and moving terms: \[ r^2 \, \sin^2(\theta) = 2r \, \cos(\theta) \]Divide both sides by \(r\) (assuming \(r eq 0\)): \[ r \, \sin^2(\theta) = 2 \, \cos(\theta) \]
04

Rewrite the final polar equation

Rewrite the equation to show the final polar form: \[ r = \frac{2 \, \cos(\theta)}{\sin^2(\theta)} \] Simplify further, if possible.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rectangular to Polar Conversion
When converting coordinates from rectangular (Cartesian) to polar form, it is essential to understand the relationship between these two systems. In rectangular coordinates, a point is defined by \(x\) and \(y\) coordinates, which represent horizontal and vertical distances from the origin, respectively. In polar coordinates, a point is defined by its distance from the origin (\(r\)) and the angle (\(\theta\)) it makes with the positive x-axis. To perform the conversion, use the formulas: \[ x = r \, \cos(\theta) \] \[ y = r \, \sin(\theta) \] Substituting these into the given rectangular equation allows us to convert it into polar coordinates.
Polar Equations
Polar equations are equations that use polar coordinates. They describe a relationship between the distance from the origin and the angle. In the exercise, we start with the rectangular equation \( y^{2}=2x \). Using the conversion formulas, we replace \(x\) and \(y\) with their polar equivalents: \[ (r \, \sin(\theta))^2 = 2 (r \, \cos(\theta)) \] Simplifying, we get: \[ r^2 \, \sin^2(\theta) = 2r \, \cos(\theta) \] After dividing by \(r\) (assuming \(r e 0\)), we arrive at: \[ r \, \sin^2(\theta) = 2 \, \cos(\theta) \] The last step is to express this as: \[ r = \frac{2 \, \cos(\theta)}{\sin^2(\theta)} \] This is the polar form of the original rectangular equation.
Trigonometric Identities
Trigonometric identities are crucial when working through problems involving polar equations. They help simplify expressions and solve equations. Common trigonometric identities include:
  • Pythagorean identity: \( \sin^2(\theta) + \cos^2(\theta) = 1 \)
  • Quotient identities: \[ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \]
  • Reciprocal identities: \[ \sec(\theta) = \frac{1}{\cos(\theta)} \]
In the exercise, using \( \sin(\theta) \) and \( \cos(\theta)\) to change from rectangular to polar forms relies on these identities. Mastery of these identities is essential to manipulate expressions and derive accurate results.

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Most popular questions from this chapter

Find the product and quotient of each pair of complex numbers using trigonometric form. Write your answers in \(a+\) bi form. $$ z_{1}=2.5+2.5 i, z_{2}=-3-3 i $$

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