Problem 4 - Thomas Calculus

Question

Identify the symmetries of the curves. Then sketch the curves in the \(x y\) -plane. \(r=1+\sin \theta\)

Step-by-Step Solution

Answer

The curve is a cardioid symmetric about the vertical line \(\theta=\frac{\pi}{2}\). Sketch by plotting key points.

Step 1: Understanding Polar Coordinates

The equation given is in polar coordinates, where each point on the curve is defined by a radius \( r \) and an angle \( \theta \). In this case, \( r = 1 + \sin \theta \).

Step 2: Analyzing Symmetry in Polar Curves

To determine symmetry, we check three types:- **Symmetry about the x-axis (horizontal axis):** Substitute \( \theta \) with \(-\theta\) and see if the equation remains unchanged.- **Symmetry about the y-axis (vertical line \(\theta = \frac{\pi}{2}\)):** Replace \( \theta \) with \(\pi - \theta\).- **Symmetry about the origin:** Apply \( r \) to \(-r \).For \( r = 1 + \sin \theta \):- Substituting \( \theta \) with \(-\theta\) gives \( r = 1 - \sin \theta \), which is not the same, so no x-axis symmetry.- Substituting \( \theta \) with \(\pi - \theta\) results in \( r = 1 + \sin \theta \), unchanged, so there is symmetry about the y-axis (vertical line at \(\theta = \frac{\pi}{2}\)).- Using \(-r\), we see it does not hold true for origin symmetry.

Step 3: Sketching the Polar Curve

To sketch \( r = 1 + \sin \theta \):- **Set \( \theta = 0 \):** \( r = 1 + \sin 0 = 1 \).- **Set \( \theta = \frac{\pi}{2} \):** \( r = 1 + \sin \frac{\pi}{2} = 2 \).- **Set \( \theta = \pi \):** \( r = 1 + \sin \pi = 1 \).- **Set \( \theta = \frac{3\pi}{2} \):** \( r = 1 + \sin \frac{3\pi}{2} = 0 \).- Plot these values and others between them to sketch the curve. The resulting shape is a cardioid centered at (1,0) with its peak point at radius 2 when \(\theta = \frac{\pi}{2}\).

Step 4: Confirming the Shape of the Curve

The shape is confirmed as a cardioid, which can be visualized by the points plotted from the sketch, verifying the symmetry about the y-axis as expected. The curve touches the origin and then loops around with a maximum radial length of 2 at \(\theta = \frac{\pi}{2}\).

Key Concepts

Symmetry

In polar coordinates, investigating symmetry helps us understand how a curve behaves when reflected across specific axes or when considering transformations. Symmetry can simplify graphing and analysis of the curves. For the polar equation \( r = 1 + \sin \theta \),:

• Check x-axis symmetry: We substitute \( \theta \) with \( -\theta \). Our equation becomes \( r = 1 - \sin \theta \), which does not match \( r = 1 + \sin \theta \). Thus, there's no x-axis symmetry.
• Check y-axis symmetry: Replace \( \theta \) with \( \pi - \theta \). The equation remains \( r = 1 + \sin \theta \), unchanged, indicating symmetry about the vertical line \( \theta = \frac{\pi}{2} \).
• Check origin symmetry: When we consider \(-r\) and \(-\theta\), the equation does not hold. Thus, there's no origin symmetry.

When sketching, recognizing y-axis symmetry can make it easier, since we only need to determine the curve's shape on one side of this axis.

Cardioid

A cardioid is a special type of heart-shaped curve, usually derived from a simple polar equation. For the equation \( r = 1 + \sin \theta \), the curve forms a classic cardioid shape where:

• Generating the shape: The curve is centered at \((1,0)\) in the polar plane. As \( \theta = 0 \), \( r \) starts at 1 and increases to a maximum of 2 as \( \theta \) reaches \( \frac{\pi}{2} \).
• Peculiar properties: The shape touches the origin at \( \theta = \frac{3\pi}{2} \) and has a consistent (though not constantly growing) distance from the origin, creating the recognizable heart-like form.
• Visual confirmation: When curves are plotted from various angles, the characteristic loop and the rounded tip of the cardioid are evident.

Understanding cardioids is integral in polar graphing, showcasing symmetrical, naturally occurring shapes.

Curve Sketching

Sketching curves in polar coordinates is a valuable skill to visualize equations and their transformations. To properly sketch the cardioid \( r = 1 + \sin \theta \), follow these steps:

• Identify key points: Calculate \( r \) at important angles such as \( \theta = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2} \). These provide anchor points to construct the curve shape systematically.
• Consider incremental angles: Evaluate \( r \) at several points between each key angle to gain a smooth, accurate representation of the curve.
• Utilize symmetry: Use symmetry about the y-axis for more efficient sketching on one side of the axis, reflecting the pattern on the opposite side.

When you plot these calculations on a coordinate plane, the sketch not only represents the cardioid's shape but also enhances comprehension of transformations in polar terms. Curve sketching involves understanding how different values of \( \theta \) affect the radius \( r \), leading to smoother diagrams.