Problem 72 - Precalculus

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 equation is \(r=4\).

Step 1: Identify the rectangular equation and the polar coordinate relationships

Given an equation in rectangular coordinates \(x^{2} + y^{2} = 16\) and the relationships between rectangular and polar coordinates: \(x = r \cos(\theta)\), \(y=r \sin(\theta)\) and \(r^2 = x^2 + y^2\).

Step 2: Substitute the rectangular coordinates with polar coordinates

Apply the relationship \(r^2 = x^2 + y^2\) to the given rectangular equation. Hence, our equation will be \(r^{2} = 16\).

Step 3: Find the polar equation

The polar equation obtained from the given rectangular equation is \(r^{2} = 16\). It can also be written as \(r = \pm 4\), indicating that the circle is traced out twice as \( \theta \) varies from \(0\) to \(2\pi\). However, it is more common to use the positive value of r, hence we can simply write \(r=4\).

Key Concepts

Rectangular Coordinates

In mathematics, rectangular coordinates, also known as Cartesian coordinates, provide a system for identifying specific positions on a plane using a pair of numerical values. These values are traditionally labeled as \((x, y)\). Each point within this coordinate system can be located on a two-dimensional grid. The \(x\)-value indicates the horizontal position, and the \(y\)-value shows the vertical position. The intersection point of the two axes is known as the origin, typically identified as \((0, 0)\).

When working with rectangular coordinates, it is essential to understand how they describe geometric shapes. For example, the equation \(x^2 + y^2 = 16\) depicts a circle with a center at the origin. This expression defines that for every point \((x, y)\) on the circle, the sum of the squares of \(x\) and \(y\) will always equal 16. Recognizing these equations can greatly aid in graphing and understanding their geometric interpretation.

Conversion to Polar Form

Converting equations from rectangular to polar form can be a valuable method for simplifying expressions, particularly when dealing with circles and other radially symmetric shapes. To perform this conversion, we rely on the relationships between rectangular and polar coordinates:

• \(x = r \cos(\theta)\)
• \(y = r \sin(\theta)\)
• \(r^2 = x^2 + y^2\)

In the given example, the original rectangular equation is \(x^2 + y^2 = 16\). By applying the relationship \(r^2 = x^2 + y^2\), we substitute \(r^2\) into the equation, leading to \(r^2 = 16\). This represents the equation in polar coordinates.

The expression \(r = \pm 4\) further simplifies the equation, showing that in polar coordinates, the circle can be described entirely by the radius. Typically, the positive radius is used, as circles are often drawn using positive values of \(r\). Hence, \(r = 4\) offers a clear and straightforward description of the circle in polar form.

Equation of a Circle

The equation of a circle is a fundamental concept in geometry that defines a perfect round shape in a coordinate system. In rectangular form, the equation of a circle with a center at the origin and radius \(r\) is given by \(x^2 + y^2 = r^2\). This simple yet powerful equation states that any point \((x, y)\) lying on the circle will satisfy the equation due to the fixed distance (radius) from the center.

In our example, the rectangular equation \(x^2 + y^2 = 16\) represents a circle with a radius of 4. This is because \(16\) is a square of 4 (\(r^2 = 16\)). Converting this to polar form, as discussed earlier, we observe that the circle can be succinctly expressed using \(r = 4\), where \(r\) continuously equals 4 as \(\theta\) varies from \(0\) to \(2\pi\).

Understanding these forms and conversions is immensely useful in solving geometry problems, facilitating easier graphing or calculus applications, and enriching one's mathematical knowledge on describing geometric figures in different systems.