Problem 71 - Algebra and Trigonometry

Question

The letters \(x\) and \(y\) represent rectangular coordinates. Write each equation using polar coordinates \((r, \theta) .\) $$ 2 x^{2}+2 y^{2}=3 $$

Step-by-Step Solution

Answer

Using polar coordinates, the equation \[ 2x^2 + 2y^2 = 3 \ becomes \[ r = \sqrt{\frac{3}{2}} \].

Step 1 - Recall the relationship between rectangular and polar coordinates

The rectangular coordinates \( x \) and \( y \) can be converted to polar coordinates \( r \) and \( \theta \) using the following relationships: \( x = r\cos(\theta) \) and \( y = r\sin(\theta) \). Also, remember that \[ r^2 = x^2 + y^2 \]

Step 2 - Replace \( x \) and \( y \) with their polar coordinate equivalents

In the given equation \[ 2x^2 + 2y^2 = 3 \], replace \( x \) and \( y \) using the relationships: \[ 2(r \cos(\theta))^2 + 2(r \sin(\theta))^2 = 3 \]

Step 3 - Simplify the equation using trigonometric identities

Use the trigonometric identities: \( \cos^2(\theta) + \sin^2(\theta) = 1 \). Simplifying the given equation we get: \[ 2r^2 (\cos^2(\theta) + \sin^2(\theta)) = 3 \]

Step 4 - Solve for \( r \)

Since \( \cos^2(\theta) + \sin^2(\theta) = 1 \), the equation simplifies to: \[ 2r^2 = 3 \. \] Divide both sides by 2: \[ r^2 = \frac{3}{2} \]

Step 5 - Take the square root to find \( r \)

Taking the square root of both sides: \[ r = \sqrt{\frac{3}{2}} \]

Key Concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, use the \((x, y)\) system to define positions on a plane. In this system, each point is described by how far it is from two fixed perpendicular lines, called the x-axis and the y-axis. The values of \(x\) and \(y\) tell you how far the point is horizontally and vertically from the origin, which is where the axes intersect.
Rectangular coordinates are great for many applications, including graphing equations and plotting data. However, they may not always be the easiest system to use for every problem. This is where other coordinate systems, like polar coordinates, can be helpful.

Polar Coordinates

Polar coordinates offer another way to determine a point's location. Instead of using \(x\) and \(y\) values, polar coordinates use \(r\) and \(\theta\).
Here, \(r\) is the distance from the origin to the point, and \(\theta\) is the angle from the positive x-axis to the line connecting the origin to the point. Think of it like finding a location based on how far and in which direction you need to go from a starting point.
This system is particularly useful for problems involving circles or any situation where angles are essential. For instance, the equation of a circle centered at the origin with radius \(r\) is simply \(r = \text{constant}\) in polar coordinates.

Trigonometric Identities

Trigonometric identities are mathematical relationships that involve trigonometric functions like sine and cosine. One essential identity is:
\[\cos^2(\theta) + \sin^2(\theta) = 1\]
This identity is incredibly useful for simplifying equations, especially those that arise when converting between rectangular and polar coordinates. In our step-by-step solution, we used it to simplify the expression involving \(r \cos(\theta)\) and \(r \sin(\theta)\). By recognizing that \(\cos^2(\theta) + \sin^2(\theta)\) equals 1, we were able to reduce the equation and solve for \(r\).

Coordinate Conversion Formulas

To convert between rectangular and polar coordinates, we use specific formulas:
\( x = r \cos(\theta) \)
\( y = r \sin(\theta) \)
\[ r^2 = x^2 + y^2 \]
The first two formulas help you convert from polar to rectangular coordinates. They give the x and y coordinates based on the distance \(r\) and angle \(\theta\).
The third formula helps convert from rectangular to polar coordinates. It provides the value of \(r\) based on the x and y coordinates.
In our exercise, we used these formulas to replace \(x^2\) and \(y^2\) with their polar equivalents and then simplified the resulting equation.