Problem 7 - Thomas Calculus

Question

Identify the symmetries of the curves. Then sketch the curves in the \(x y\) -plane. \(r=\sin (\theta / 2)\)

Step-by-Step Solution

Answer

The curve is symmetric about the x-axis and resembles a heart-like shape.

Step 1: Convert to Cartesian Coordinates

The given polar equation is \( r = \sin(\theta/2) \). To analyze symmetry, let's first express it in Cartesian coordinates. Recall the polar to Cartesian conversions: \( x = r\cos\theta \) and \( y = r\sin\theta \). Using the identity \( \sin\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 - \cos\theta}{2}} \), we can write the equation in terms of \( x \) and \( y \). However, it's easier to directly consider symmetry properties in polar form.

Step 2: Analyze Symmetry with respect to the Polar Axis

For symmetry concerning the polar axis (x-axis), if \( r = f(\theta) \) is a solution, then \( r = -f(-\theta) \) is also a solution. Here, substituting \( -\theta/2 \) gives \( r = \sin(-\theta/2) = -\sin(\theta/2) \). Due to the sine function's odd nature, the equation exhibits symmetry about the x-axis.

Step 3: Check for Symmetry with respect to the Line \(\theta = \frac{\pi}{2}\)

If the curve is symmetric about \(\theta = \frac{\pi}{2}\), then replacing \( \theta \) by \( \pi - \theta \) should give the same equation. Here, \( r = \sin((\pi - \theta)/2) = \sin((\pi/2) - \theta/2) = \cos(\theta/2) \), which does not equal \( \sin(\theta/2) \). Hence, there is no symmetry about the line \(\theta = \frac{\pi}{2}\).

Step 4: Check for Symmetry with respect to the Pole

Symmetry with respect to the pole (origin) means replacing \( r \) by \( -r \), giving \( r = -\sin(\theta/2) \). Since \( \sin(\theta/2) \) becomes zero at multiples of \(\pi\), there is no symmetry about the pole, except at these points, which are inherently symmetric.

Step 5: Sketch the Curve

The curve is a limacon with the inner loop absent because of symmetry about the x-axis. The values of \( \theta \) range from \(0\) to \(2\pi\). As \(\theta\) progresses, the values of \(r\) first increase, peaking at \(\theta = \pi\), and then decreasing symmetrically.Sketching it from \( \theta = 0 \) to \( 2\pi \), we see a single arch beginning and ending at the pole, creating a heart-like shape.

Key Concepts

Symmetry in Polar Graphs

Symmetry in polar graphs is a fascinating study area because it highlights different properties of the curve based on the equation's format. Identifying symmetry can simplify the graphing process by showing repeated patterns without needing to plot every point.
For curves represented in polar coordinates, symmetry can happen in three main ways:

• Polar axis symmetry: This is similar to symmetry about the x-axis in Cartesian coordinates. A polar graph is symmetric about the polar axis if replacing \( (r, \theta) \) with \( (r, -\theta) \) gives the same equation.
• Line \( \theta = \frac{\pi}{2} \) symmetry: Similar to the y-axis symmetry in the Cartesian plane, it occurs if the graph remains unchanged when \( \theta \) is replaced by \( \pi - \theta \). In the exercise, the lack of symmetry shows up because transformations do not yield an equivalent expression.
• Pole symmetry: This resembles origin symmetry in Cartesian graphs. It appears if \((r, \theta)\) becomes \((-r, \theta)\) results in the same equation, indicating the pole is a central point of symmetry.

Understanding these symmetries can provide insight into the shape and characteristics of polar curves, leading to more efficient sketching without trial and error.

Cartesian Coordinates Conversion

Converting polar coordinates into Cartesian coordinates involves a set of equations that bridge the polar and Cartesian systems:

• X-Coordinate: \( x = r\cos\theta \)
• Y-Coordinate: \( y = r\sin\theta \)

These formulas derive from the geometrical relationship within a right-angled triangle formed by a radius line at an angle \( \theta \), showing how a point in polar form translates into one defined by x and y. This conversion is useful for many tasks, such as analyzing curves and solving intersection problems in a familiar Cartesian plane.
This step-by-step transformation provides clear algebraic expressions to work with while maintaining contextual awareness of polar graphs' unique attributes. While some symmetry aspects are better investigated in their original polar format, translating into Cartesian coordinates can simplify complex expressions, allowing for a broader analysis with established Cartesian methods.

Limacon Curves

Limacon curves, often characterized by their heart-like shape, are particularly interesting polar graphs resulting from a form like \( r = a + b\sin\theta \) or \( r = a + b\cos\theta \):

• These curves can feature an inner loop, but this depends greatly on the values of parameters \( a \) and \( b \).
• When \( |b| > |a| \), the graph typically shows a distinct inner loop.
• If \( |b| = |a| \), the curve forms a cardioid, an eye-catching figure with a cusp at the pole.
• In scenarios where \( |a| > |b| \), the inner loop vanishes, leaving a dimpled, smooth outer curve.

In our original problem, the lack of necessity for conversion to a cardioid indicates it's a limacon without an inner loop. Understanding limacon curves provides a foundation for exploring more complex polar curves. Such knowledge aids in graphing, analyzing real-world applications, and recognizing inherent beauty in mathematical forms.