Problem 48 - Algebra and Trigonometry Real Mathematics, Real People

Question

Convert the rectangular equation to polar form. Assume \(a<0\) $$x^{2}+y^{2}=16$$

Step-by-Step Solution

Answer

The polar form of the given rectangular equation \(x^{2}+y^{2}=16\) is \(r^{2}=16\).

Step 1: Write down the Rectangular Equation

We have the rectangular equation \(x^{2}+y^{2}=16\).

Step 2: Apply Polar Conversion

To convert the given equation to polar form, replace \(x\) and \(y\) by their polar form representations. That is, replace \(x\) with \(rcos(\theta)\) and \(y\) with \(rsin(\theta)\). Doing so gives us \((rcos(\theta))^{2} + (rsin(\theta))^{2}=16.\)

Step 3: Simplify the Equation

The expression becomes \((rcos(\theta))^{2} + (rsin(\theta))^{2}\) simplifies to \(r^{2}(cos^{2}(\theta) + sin^{2}(\theta)) = 16\). But we know that \(cos^2(\theta) + sin^2(\theta) = 1\) for any \(\theta\). Therefore the final equation simplifies to \(r^{2}=16\).

Key Concepts

Rectangular Coordinates

In mathematics, rectangular coordinates are part of the Cartesian coordinate system, which uses two or three numerical values to determine a point's position in a 2D or 3D space. For simplicity, we'll focus on 2D space, where these coordinates are expressed as \( (x, y) \).

• The \( x \)-value represents the horizontal position, while the \( y \)-value indicates the vertical position.
• These values come together to pinpoint the location of a point on a grid.

The equation presented in the Original Exercise \( x^2 + y^2 = 16 \) demonstrates a circle with a radius of 4 units centered at the origin. In this equation, both \( x \) and \( y \) vary but adhere to the required sum of their squares, which is equal to the constant in the equation.

Coordinate Conversion

Coordinate conversion involves changing the representation of the coordinates of any geometric figure from one system to another. In the problem you've encountered, you're converting from rectangular coordinates \( (x, y) \) to polar coordinates \( (r, \theta) \). These two systems describe the same point but with different parameters.
To convert:

• Use the formulas \( x = r \cos(\theta) \) and \( y = r \sin(\theta) \).
• The distance \( r \) from the origin is computed by \( r = \sqrt{x^2 + y^2} \).
• The angle \( \theta \) subtended at the origin is given by \( \theta = \tan^{-1}(\frac{y}{x}) \).

Applying these conversions in the original equation \( x^2 + y^2 = 16 \), the polar form comes to life as \( r^2 = 16 \). Note here how we eliminated \( \theta \) due to the use of trigonometric identities.

Trigonometric Identities

Trigonometric identities are mathematical equations that involve trigonometric functions and are true for every value of the occurring variables where both sides of the identity are defined. They play a crucial role in simplifying expressions and solving equations as evidenced in the solution of the exercise.
A key identity we've employed here is the Pythagorean identity:

• \[ \cos^2(\theta) + \sin^2(\theta) = 1 \]

This identity allows you to simplify the equation \( r^2(\cos^2(\theta) + \sin^2(\theta)) = 16 \) to \( r^2 = 16 \), making the conversion from rectangular to polar form seamless.
Understanding these identities not only helps in coordinate conversion but also underpins solutions to countless mathematical and physical problems, making them essential tools in any mathematician's kit.