Problem 21 - Precalculus : Building Concepts and Connections

Question

In Exercises \(7-22,\) sketch the graphs of the polar equations. $$r=\frac{3}{2} \cos \theta$$

Step-by-Step Solution

Answer

The graph represents a circle centered at the origin with radius \(\frac{3}{2}\). It starts at \(\theta=0\) where \(r=\frac{3}{2}\). As \(\theta\) increases, \(r\) decreases until \(\theta = 180^\circ\), then \(r\) increases again, coming back to the initial starting point.

Step 1: Understanding the polar coordinates

A polar coordinate system involves distance and angle. To sketch polar coordinates graphically, we need to understand that \(r\) represents the radial distance from the origin, and \(\theta\) refers to the counter-clockwise angle from the positive x-axis.

Step 2: Convert the polar equation to Cartesian coordinates

In polar coordinates, we can convert to Cartesian using the following relationships: \(x=r\cos\theta\) and \(y=r\sin\theta\). So, substituting \(r = \frac{3}{2}\cos\theta\) into the equations, we get \(x = r\cos\theta = (\frac{3}{2}\cos\theta)\cos\theta= \frac{3}{2}\cos^2\theta\) and \(y = r\sin\theta = (\frac{3}{2}\cos\theta)\sin\theta\). The trigonometric identity \(\cos^2\theta + \sin^2\theta = 1\) can simplify this to \(x = r = \frac{3}{2}\cos\theta\) and \(y = \sqrt{r^2 - x^2}\). Also notice that \(r = \frac{3}{2}\cos\theta\) is a circle with radius \(\frac{3}{2}\) in Cartesian coordinates.

Step 3: Plot the graph

Start at \(\theta=0\) where \(r=\frac{3}{2}\), which will be your maximum. This means we start on the positive x-axis \(\frac{3}{2}\) units away from the origin. As \(\theta\) increases, \(r\) decreases until \(\theta = 180^\circ\). Then, \(r\) increases again, coming back to the initial starting point. The plot will represent a circle centered at the origin with radius \(\frac{3}{2}\).

Key Concepts

Polar Coordinates

When we encounter equations and graphs in mathematics, we often default to the Cartesian coordinate system, which uses horizontal and vertical axes to denote location with x and y coordinates. However, the polar coordinate system offers a distinctive way of representing points in the plane through the use of distance and angles relative to a central point, also known as the pole (commonly the origin).

In polar coordinates, a point is represented by a pair \( (r, \theta) \) where \(r\) is the radial distance from the origin and \(\theta\) is the angle made with the positive x-axis, measured in a counter-clockwise direction. Unlike the Cartesian system, which describes location via a rectangle's sides, the polar system uses a segment and an angle, bringing a different perspective, especially when dealing with curves and shapes that are circular or exhibit radial symmetry.

For instance, when describing a circle, the polar form is often more straightforward. As the angle \(\theta\) sweeps around the origin, the radius \(r\) may remain constant, immediately describing a circle. This simple representation streamlines the process of analyzing and sketching geometries like spirals or roses that are inherently based on radial distances and angles.

Converting Polar to Cartesian Coordinates

To bridge the gap between the polar and Cartesian systems, we can employ conversion formulas. Understanding these transformations is crucial as it can enable us to switch from polar to Cartesian coordinates effortlessly, providing a connection between the two systems and offering flexibility in solving problems.

The transformation is straightforward, using trigonometric relationships: \(x = r\cos(\theta)\) and \(y = r\sin(\theta)\). By plugging the polar coordinates into these equations, we translate the polar point \( (r, \theta) \) into its Cartesian counterpart \( (x, y) \) by projecting the point onto the x and y axes.

For example, if we have a polar equation like \( r = \frac{3}{2}\cos(\theta) \), we convert this to Cartesian form by substituting \(r\) into the conversion formulas, resulting in \( x = \frac{3}{2}\cos^2(\theta) \) and \( y = \frac{3}{2}\cos(\theta)\sin(\theta) \). This allows us to interpret and sketch the equation's graph in the Cartesian plane, typically a useful step when working with certain integrals, differential equations, or optimizing problems where Cartesian coordinates are more practical or required.

Trigonometric Identities

Understanding trigonometric identities is akin to having a powerful toolset at your disposal. These mathematical facts express relationships between trigonometric functions and are instrumental in simplifying expressions, solving equations, and converting between different forms of representation.

A fundamental identity is \( \cos^2(\theta) + \sin^2(\theta) = 1 \), which arises from the Pythagorean theorem applied to a unit circle. This identity allows us to relate the Cartesian coordinates \(x\) and \(y\) back to the radius \(r\) of polar coordinates, especially when we convert polar equations into Cartesian form. It also enables us to rewrite trigonometric expressions in different forms, making certain calculations more manageable.

For instance, in the context of converting polar equations, we can harness this identity to manipulate the expressions for \(x\) and \(y\) into more familiar forms. By recognizing that \(\cos^2(\theta)\) in the expression for \(x\) is inherently linked to \(r\) given the \(r^2 = x^2 + y^2\) relationship in a circle, we can often simplify these equations substantially, revealing underlying shapes or facilitating further computation. Employing trigonometric identities is a crucial step in both simplifying and understanding the expressions we work with in mathematics.