Problem 50 - Precalculus Mathematics for Calculus

Question

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). $$r=\sin \theta+2 \cos \theta$$

Step-by-Step Solution

Answer

The parametric equations are \( x=\sin\theta \cos\theta + 2 \cos^2 \theta \) and \( y=\sin^2\theta + 2 \cos \theta \sin \theta \).

Step 1: Understand the Polar Equation

The given polar equation is \( r = \sin \theta + 2 \cos \theta \). This equation gives \( r \) as a function of \( \theta \), where \( r \) is the radial distance from the origin and \( \theta \) is the angle from the positive x-axis.

Step 2: Use Parametric Equations for Polar Coordinates

In polar coordinates, the relationship between \( x \), \( y \), \( r \), and \( \theta \) is defined by \( x = r \cos \theta \) and \( y = r \sin \theta \). We can use these expressions to convert \( r = \sin \theta + 2 \cos \theta \) to parametric form.

Step 3: Express Parametric Equations

Substitute \( r = \sin \theta + 2 \cos \theta \) into the parametric equations: - \( x = r \cos \theta = (\sin \theta + 2 \cos \theta) \cos \theta \),- \( y = r \sin \theta = (\sin \theta + 2 \cos \theta) \sin \theta \).These become the parametric equations: - \( x = \sin \theta \cos \theta + 2 \cos^2 \theta \),- \( y = \sin^2 \theta + 2 \cos \theta \sin \theta \).

Step 4: Rearrange and Simplify Parametric Equations

Further simplify the parametric equations: - Substitute \( \sin^2 \theta = 1 - \cos^2 \theta \) for simplification if needed,- However here main focus is finding an immediate form, so using trigonometric identities: - \( x = \sin \theta \cos \theta + 2 \cos^2 \theta \) - \( y = \sin^2 \theta + 2 \cos \theta \sin \theta \)

Step 5: Graph the Parametric Equations

To graph the resulting parametric equations, use a graphing tool or software. Enter the parametric form: - \( x = \sin \theta \cos \theta + 2 \cos^2 \theta \)- \( y = \sin^2 \theta + 2 \cos \theta \sin \theta \)Plot the curve for values of \( \theta \) from 0 to \( 2\pi \) to see a complete representation of the graph.

Key Concepts

Parametric Equations

Parametric equations are a way of expressing a set of related quantities as explicit functions of an independent parameter, usually denoted by \( t \). This method of defining equations is particularly useful in describing motion or paths in coordinate spaces. In the context of polar coordinates, parametric equations allow us to translate angles and distances into Cartesian coordinates, making them essential for graphing and further analysis.
For example, consider the polar equation \( r = \sin \theta + 2 \cos \theta \). To convert this into parametric form, we use the relationships \( x = r \cos \theta \) and \( y = r \sin \theta \). This conversion:

• Helps express complex shapes more comprehensively.
• Makes equations easier to interpret in graphing scenarios.

Thus, by substituting our original polar equation into these formulas, we obtain a pair of related parametric equations for \( x \) and \( y \), which can then be graphed or analyzed using various mathematical tools.

Polar Equation Conversion

Converting polar equations into other forms is vital for simplifying mathematical expressions and enhancing their utility. Such conversions allow the representation of polar equations in Cartesian (\( x, y \)) terms, which are easier to handle for most analytical and graphing purposes.
When dealing with the equation \( r = \sin \theta + 2 \cos \theta \), the conversion process involves making substitutions based on the trigonometric identities and relationships between polar and Cartesian coordinates:

• \( r \) can be substituted into \( x = r \cos \theta \) and \( y = r \sin \theta \).
• By simplifying using trigonometric identities like \( \sin^2 \theta + \cos^2 \theta = 1 \), more readable forms of the equations are derived.

These conversions allow us to visualize polar equations by plotting them through easily accessible parametric form. The flexibility of this transformation also helps in mathematical exploration and discovering properties of the curve.

Graphing Polar Equations

Graphing polar equations can provide insightful visual representations that are often not as apparent with the raw equations. The challenge lies in transforming polar coordinates into a form that graphing software or devices can understand easily.
Given the parametric equations obtained from \( r = \sin \theta + 2 \cos \theta \), the plotting process involves:

• Defining the range for \( \theta \), typically from 0 to \( 2\pi \), to visualize the complete curve.
• Using the parametric form, \( x = \sin \theta \cos \theta + 2 \cos^2 \theta \) and \( y = \sin^2 \theta + 2 \cos \theta \sin \theta \), to input into graphing software.

This method of graphing reveals the geometrical nature of the equation, often unveiling symmetrical properties and dynamic changes in the curve as \( \theta \) varies. Such insights are crucial in subjects like physics and engineering, where understanding the movement and shape is essential.