Problem 95 - Precalculus

Question

The use of a different system of coordinates simplifies many mathematical expressions and some calculations are performed in an easier way. The rectangular coordinates \((x, y)\) are transformed into polar coordinates by the equations $$x=r \cos \theta \quad \text { and } \quad y=r \sin \theta$$ where \(r=\sqrt{x^{2}+y^{2}}(r \neq 0)\) and \(\tan \theta=\frac{y}{x}(x \neq 0) .\) In polar coordinates, the equation of the unit circle \(x^{2}+y^{2}=1\) is just \(r=1\) In calculus, we use polar coordinates extensively. Transform the rectangular equation to polar form. $$x^{2}+y^{2}=25$$

Step-by-Step Solution

Answer

The polar form of the equation is \(r = 5\).

Step 1: Identify the Original Equation

The original equation given in rectangular coordinates is the equation of a circle centered at the origin: \(x^2 + y^2 = 25\).

Step 2: Recall the Polar Coordinate Formulas

In polar coordinates, \(x = r \cos \theta\) and \(y = r \sin \theta\). Additionally, we have \(r = \sqrt{x^2 + y^2}\).

Step 3: Substitute into the Equation

Substitute the expressions for \(x\) and \(y\) in terms of \(r\) and \(\theta\) into the original equation. This gives: \((r \cos \theta)^2 + (r \sin \theta)^2 = 25\).

Step 4: Simplify the Equation

Notice that \((r \cos \theta)^2 + (r \sin \theta)^2 = r^2 (\cos^2 \theta + \sin^2 \theta)\). Since \(\cos^2 \theta + \sin^2 \theta = 1\), the equation simplifies to \(r^2 = 25\).

Step 5: Solve for r

Take the square root of both sides to solve for \(r\). Thus, \(r = \sqrt{25}\) or \(r = 5\).

Step 6: Final Polar Form

The equation in polar coordinates is \(r = 5\).

Key Concepts

Rectangular Coordinates

In mathematics and geometry, we often start learning about rectangular coordinates, which are also called Cartesian coordinates. This system uses two values to determine a precise position of a point on a 2D plane. The values are represented as \((x, y)\).
- **X-coordinate**: This first number tells us how far left or right the point is from the y-axis.
- **Y-coordinate**: This second number tells us how far up or down the point is from the x-axis.
These coordinates are incredibly useful in graphing equations, as they allow us to plot points on a graph and understand geometric shapes like lines, circles, and parabolas. The purpose of converting these coordinates into other forms, like polar coordinates, is often to simplify calculations and shed new theoretical light on familiar geometric problems.

Coordinate Transformation

Coordinate transformation is a key concept that involves changing from one coordinate system to another. This is particularly useful when solving complex equations or dealing with specific geometric shapes where one coordinate system may be more advantageous than another.
- **Polar Coordinates**: A common transformation is between rectangular and polar coordinates. In polar coordinates, a point in the plane is represented by \((r, \theta)\), where \(r\) is the distance from the origin, and \(\theta\) is the angle from the positive x-axis.
To convert from rectangular to polar:

• \(x = r \cos \theta\)


• \(y = r \sin \theta\)


• Use \(r = \sqrt{x^2 + y^2}\)


• \(\tan \theta = \frac{y}{x}\)

Such transformations often reveal new insights about shapes and simplify trigonometric or calculus problems.

Equation of a Circle

Understanding the equation of a circle is a fundamental aspect of geometry. The standard form of a circle's equation in rectangular coordinates is \(x^2 + y^2 = r^2\), where \((x, y)\) are the coordinates of any point on the circle, and \(r\) is the radius of the circle. Now, after transforming to polar coordinates, this equation simplifies significantly.
For a circle centered at the origin, the polar form becomes simply \(r = \text{constant}\), which is the circle's radius. This is because:

• The expression \(\cos^2 \theta + \sin^2 \theta = 1\) is always true for any angle \(\theta\).


• This reduces the circle's equation to primarily focus on \(r\), the radius.

This simplification highlights the power of polar coordinates, especially in calculating with trigonometric integrals or understanding symmetry and rotational properties.