Problem 51 - Algebra and Trigonometry

Question

\(49-52\) . A polar equation is given. (a) Express the polar equation in parametric form. (b) Use a graphing device to graph the parametric equations you found in part (a). $$ r=\frac{4}{2-\cos \theta} $$

Step-by-Step Solution

Answer

Express as parametric: \(x = \frac{4\cos\theta}{2-\cos\theta}\), \(y = \frac{4\sin\theta}{2-\cos\theta}\). Graph these for visualization.

Step 1: Understand the Polar Equation

We are given the polar equation \(r = \frac{4}{2 - \cos \theta}\). A polar equation relates the distance from the origin \(r\) to the angle \(\theta\). Our task is to express this in parametric form, typically using \(x\) and \(y\) coordinates.

Step 2: Express \(r\) in Terms of \(x\) and \(y\)

Recall the relations in polar coordinates where \(x = r\cos\theta\) and \(y = r\sin\theta\). We solve for \(r\) in terms of cosine and sine functions.

Step 3: Substitute the Expression for \(r\)

Using \(r = \frac{4}{2 - \cos\theta}\), substitute this into the relations for \(x\) and \(y\): \[x = \left(\frac{4}{2-\cos\theta}\right)\cos\theta, \quad y = \left(\frac{4}{2-\cos \theta}\right)\sin \theta\]

Step 4: Simplify the Parametric Equations

Further simplify:\(x = \frac{4\cos\theta}{2-\cos\theta}\) and \(y = \frac{4\sin\theta}{2-\cos\theta}\). We now have the parametric form of the original polar equation.

Step 5: Graph the Parametric Equations

Using a graphing calculator or software, input the parametric equations \(x(t) = \frac{4\cos t}{2-\cos t}\) and \(y(t) = \frac{4\sin t}{2-\cos t}\), where \(t\) serves as the parameter representing \(\theta\). Graph these equations to visualize the curve.

Key Concepts

Parametric Equations

Parametric equations provide an alternative way to describe curves, using a parameter, often denoted as \( t \). In the context of our polar equation, the parameter \( \theta \) is utilized. Polar coordinates, which are expressed as \((r, \theta)\), can be transformed into parametric equations involving \( x \) and \( y \). This transformation is beneficial for graphing and analysis.To convert our polar equation, \( r = \frac{4}{2 - \cos \theta} \), into parametric form, we equate it to the polar coordinate relationships:- \( x = r \cos \theta = \frac{4 \cos \theta}{2 - \cos \theta} \)- \( y = r \sin \theta = \frac{4 \sin \theta}{2 - \cos \theta} \)These simplified parametric equations are crucial as they allow us to easily plot and understand the shape of the curve. The parameter \( t \) varies, often over \([0, 2\pi]\) or a similar interval, providing a dynamic view of the relationship between \( x \) and \( y \).

Graphing Polar Equations

Graphing polar equations can initially seem daunting, but converting them into parametric form simplifies the process significantly, especially with digital graphing tools. With our parametric equations \( x(t) = \frac{4 \cos t}{2 - \cos t} \) and \( y(t) = \frac{4 \sin t}{2 - \cos t} \), we can utilize graphing software to easily plot the curve.Here's how you graph these equations:- Enter the parametric equations into a graphing device or calculator.- Ensure the parameter \( t \) is within the required interval (often \([0, 2\pi]\)).- Observe the plotted graph, which provides a visual representation of the original polar equation.By graphing the parametric equations, students can better understand the spatial characteristics of the curve and how changes in \( \theta \) affect the overall shape and position of the graph. This visualization can be extremely helpful for developing a deeper understanding of polar equations and their applications.

Converting Polar to Cartesian Coordinates

Converting polar coordinates to Cartesian coordinates is a fundamental skill when dealing with curves and functions described in polar form. Polar coordinates \((r, \theta)\) communicate a point's distance from the origin and direction, while Cartesian coordinates \((x, y)\) place the point on a grid.To perform this conversion, we use the following relationships:- \( x = r \cos \theta \)- \( y = r \sin \theta \)For our specific equation, \( r = \frac{4}{2 - \cos \theta} \), we substitute \( r \) into these formulas to get:- \( x = \frac{4 \cos \theta}{2 - \cos \theta} \)- \( y = \frac{4 \sin \theta}{2 - \cos \theta} \)These equations link the polar form back to Cartesian coordinates, illustrating the practical utility of both coordinate systems. By understanding the conversion process, students can transition seamlessly between systems, using the most effective method for their particular problems.