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Chapter 6: Problem 68

Convert the rectangular equation to polar form. Assume \(a>0\). \(y=x\)

Short Answer

Expert verified
The polar form of the equation is \(胃 = 蟺/4\).

Step by step solution

01

Write the formulae for changing coordinates

To convert the given equation into polar coordinates, use the formulas \(x = r\cos 胃\) and \(y = r\sin 胃\).
02

Substituting the conversion formulae into the equation

Substitute the above expressions into the given equation \(y=x\). This gives \(r\sin 胃 = r\cos 胃\).
03

Simplify the Equation

The equation can be simplified by dividing both the sides by \(r\) and cos 胃, the equation will be simplified to \(\tan 胃 = 1\).
04

Solve for theta

The angle \(胃\) will be \(蟺/4\) when \(\tan 胃 = 1\).

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Key Concepts

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

Rectangular to Polar Conversion
Converting a rectangular equation like \( y = x \) into polar form requires understanding how both coordinate systems relate to each other. In the rectangular form, \( x \) and \( y \) are coordinates that define a position on a plane through horizontal and vertical distances, respectively. The polar system, however, uses a radius \( r \) and an angle \( \theta \) to pinpoint a location.

Here's how you can make this transformation:
  • Use the conversion formulas: \( x = r \cos \theta \) and \( y = r \sin \theta \).
  • Substitute these into the given equation: \( y = x \) becomes \( r \sin \theta = r \cos \theta \).
These steps change the Cartesian equation to one that tells us the relationship between the radius and angle, both crucial for graphing in polar coordinates.
Trigonometric Identities
Trigonometric identities play a vital role in simplifying polar equations. Once we substitute \( x = r \cos \theta \) and \( y = r \sin \theta \) into the equation \( y = x \), we reach a trigonometric equation: \( r \sin \theta = r \cos \theta \).

Here's the key idea:
  • Divide both sides by \( r \) (assuming \( r eq 0 \)) and then by \( \cos \theta \): This gives us \( \frac{\sin \theta}{\cos \theta} = 1 \).
  • Recognize that \( \frac{\sin \theta}{\cos \theta} = \tan \theta \). So, \( \tan \theta = 1 \).
From the identity \( \tan \theta = 1 \), we can solve for \( \theta \) to find that \( \theta = \frac{\pi}{4} \). This value is a standard angle where the tangent function equals one, reflecting a 45-degree angle with equal x and y components.
Coordinate Transformation
Transforming coordinates involves reinterpreting the position of a point from one system to another. This isn't just a simple swap; it requires understanding how the systems work. For our exercise, we started with a line \( y = x \) in rectangular form. By switching to polar, the same line becomes an angle, specifically \( \theta = \frac{\pi}{4} \).

Think about it this way:
  • The line \( y = x \) consists of all points where the vertical and horizontal distances are equal.
  • In polar coordinates, this translates to a path traced by continuously increasing \( r \), but maintaining the constant angle \( \theta = \frac{\pi}{4} \).
This transformation shows how realizing a curve or line can appear vastly different depending on the coordinate system used, emphasizing the elegance and utility of polar graphs for certain mathematical interpretations.

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