Problem 10 - Thomas Calculus

Question

Identify the symmetries of the curves in Exercises \(1-12 .\) Then sketch the curves. $$ r^{2}=\sin \theta $$

Step-by-Step Solution

Answer

The curve is symmetric about \(\theta = \frac{\pi}{2}\) and the pole. It resembles a double-leafed clover.

Step 1: Identify Symmetries

In polar coordinates, a curve can exhibit symmetries about the polar axis, the line \(\theta = \frac{\pi}{2}\), and the pole. A common test for polar symmetry is substituting \(-\theta\) for \(\theta\). For the given curve, substitute \(-\theta\) yielding: \[r^2 = \sin(-\theta) = -\sin(\theta)\]The equation \(r^2 = \sin(\theta)\) and \(r^2 = -\sin(\theta)\) are not equivalent, so the curve is not symmetric about the polar axis.

Step 2: Test for Symmetry about \(\theta = \frac{\pi}{2}\)

For symmetry about the line \(\theta = \frac{\pi}{2}\), substitute \(\pi - \theta\) for \(\theta\). So \(r^2 = \sin(\pi - \theta) = \sin(\theta)\).The equation remains unchanged, indicating that the curve is symmetric about the line \(\theta = \frac{\pi}{2}\).

Step 3: Test for Symmetry about the Pole

For symmetry about the pole, substitute \(-r\) for \(r\). This means:\[(-r)^2 = r^2 = \sin(\theta)\]The original equation is retained, so the curve is symmetric with respect to the pole.

Step 4: Sketch the Curve

Having identified the symmetries, sketch the curve. Ranging \(\theta\) from \(0\) to \(2\pi\):- At \(\theta = 0\), \(r^2 = \sin(0) = 0\), so \(r = 0\).- At \(\theta = \frac{\pi}{2}\), \(r^2 = \sin\left(\frac{\pi}{2}\right) = 1\), giving \(r = \pm 1\).- At \(\theta = \pi\), \(r^2 = \sin(\pi) = 0\), so \(r = 0\).- This pattern repeats for \(\theta = \frac{3\pi}{2}\) and \(\theta = 2\pi\).The curve resembles a double-leafed clover, symmetric about \(\theta = \frac{\pi}{2}\) and about the pole.

Key Concepts

Curve Symmetry

In polar coordinates, curve symmetry helps us understand how a curve behaves under certain transformations. A polar curve can display symmetry in various ways, including symmetry about the polar axis, symmetry about the line \(\theta = \frac{\pi}{2}\), and symmetry about the pole. \

• **Symmetry about the Polar Axis**: Substituting \(-\theta\) for \(\theta\) in the polar equation checks this symmetry. If the equation remains unchanged or equivalent, the curve is symmetric about the polar axis.
• **Symmetry about \(\theta = \frac{\pi}{2}\)**: Here, we replace \(\theta\) with \(\pi - \theta\). When the equation stays the same, it confirms this type of symmetry.
• **Symmetry about the Pole**: To test this, substitute \(-r\) for \(r\). If the equation still holds, the curve is symmetric about the pole.

For example, when you consider the curve \(r^2 = \sin(\theta)\), applying these tests reveals the curve’s symmetric features and aids in understanding its shape without sketching it. Also, it enables easier sketching of complex curves by knowing their symmetrical behaviors.

Polar Axis

The polar axis is a crucial concept in polar coordinates, serving as the baseline from which angles are measured. It represents a horizontal line from the origin, analogous to the positive x-axis in Cartesian coordinates. \When discussing curve symmetry, testing for symmetry about the polar axis involves checking if the curve is a mirror image across this line. To perform this test, as described earlier, replace \(\theta\) with \(-\theta\) in the given equation and observe if the equation remains unaltered. This test is fundamental because symmetry about the polar axis often means that plotting points for one part of the curve gives insight into plotting the rest, thereby simplifying the process of sketching the entire curve. By understanding the role of the polar axis, students can more effortlessly analyze polar curves and understand the nature and direction of the graph at a glance.

Sketching Polar Curves

Sketching polar curves involves plotting points based on polar coordinates, where each point in the plane is defined by a distance from the origin (\(r\)) and an angle (\(\theta\)) from the polar axis. \To sketch the curve, it's beneficial to identify key points such as: \

• **Where \(\theta = 0\)**: This determines points along the polar axis initially.
• **Where \(\theta = \frac{\pi}{2}\), \(\pi\), \(\frac{3\pi}{2}\), etc.**: These angles help identify symmetry and repetitive patterns.
• **Where \(r = 0\)**: Identifying radial symmetry or loops with respect to the pole.

Once you have a few key points, you can begin to see the shape form. For the curve \(r^2 = \sin(\theta)\), you'll notice patterns forming a leaf-like structure. This structure benefits from symmetry about the line \(\theta = \frac{\pi}{2}\) and the pole, making it appear as a double-leafed shape when sketched.

Symmetry Tests

Symmetry tests help determine how polar curves relate to each other or to a line by checking invariance under certain transformations. \Here's a quick rundown on performing these tests effectively: \

• **Polar Axis Symmetry**: Replace \(\theta\) with \(-\theta\) in the equation. If unchanged, the curve is symmetric about the polar axis.
• **\(\theta = \frac{\pi}{2}\) Symmetry**: Substitute \(\theta\) with \(\pi - \theta\). If the equation results are identical, it confirms this symmetry.
• **Pole Symmetry**: Substitute \(r\) with \(-r\). If true, you have found symmetry at the pole.

These tests are foundational for not only sketching and analyzing curves but also verifying the inherent properties of the function. They ensure accuracy in representation and interpretation, simplifying many aspects of working with polar curves.