Problem 113 - Calculus Early Transcendentals

Question

Without using a graphing utility, determine the symmetries (if any) of the curve \(r=4-\sin (\theta / 2)\).

Step-by-Step Solution

Answer

Answer: The curve does not exhibit radial or angular symmetry.

Step 1: Test for Radial Symmetry

To test for radial symmetry, we need to check whether the polar equation remains the same when we replace \(\theta\) with \(\theta + \pi\). If so, the curve is symmetric with respect to the origin. \(r(4 - \sin((\theta + \pi) / 2)) = 4 - \sin(\frac{\theta}{2} + \frac{\pi}{2})\) Now we have to check if this equation is the same as the original equation, \(r = 4 - \sin(\theta / 2)\). Notice that \(\sin(\frac{\theta}{2} + \frac{\pi}{2}) \neq \sin(\theta / 2)\). Thus, the curve is not symmetric with respect to the origin.

Step 2: Test for Angular Symmetry

To test for angular symmetry, we need to check whether the polar equation remains the same when we replace \(\theta\) with \(-\theta\). If so, the curve is symmetric with respect to the polar axis. \(r(4 - \sin(-\theta / 2)) = 4 - \sin(-\frac{\theta}{2})\) Recall that \(\sin(-x) = -\sin(x)\). Therefore, we have: \(4 - \sin(-\frac{\theta}{2}) = 4 + \sin(\frac{\theta}{2})\) Now we have to check if this equation is the same as the original equation, \(r = 4 - \sin(\theta / 2)\). Notice that \(4 + \sin(\frac{\theta}{2}) \neq 4 - \sin(\theta / 2)\). Thus, the curve is not symmetric with respect to the polar axis. Since the curve does not exhibit radial symmetry nor angular symmetry, it does not have any of the traditional polar symmetries.

Key Concepts

Symmetry

Symmetry in polar curves refers to the idea that a particular graph or curve looks the same after applying certain transformations. These transformations could be reflections or rotations. For polar equations, we particularly look at three types of symmetries:

• Radial Symmetry: This occurs if the curve remains unchanged after rotation around the pole by \( \pi \) radians. To test for radial symmetry, substitute \( \theta \, \) with \( \theta + \pi \). If the equation remains unchanged, the curve is symmetric with respect to the origin.
• Angular Symmetry: This type of symmetry is present if the equation remains the same when \( \theta \, \) is replaced with \( -\theta \). If so, the curve is symmetric with respect to the polar axis.
• Polar Symmetry: If replacing \( r \, \) with \( -r \) renders an unchanged equation, the graph shows symmetry with respect to the pole.

Understanding symmetry helps in sketching polar graphs without detailed computations or graphing utilities. By recognizing symmetries, you can infer important properties of the curve.

Polar Equations

Polar equations provide a unique method of representing curves or graphs by specifying the distance from a point termed the "pole," and the angle from a reference direction called the "polar axis." With polar equations, you express the relationship as \( r = f(\theta) \), where:

• \( r \) indicates the radius or distance from the pole to a point on the curve.
• \( \theta \) signifies the angle extending from the polar axis.

This system is ideal for depicting spirals, circles, and many other shapes efficiently. It differs from the Cartesian system, which uses \( x \) and \( y \) coordinates to define a location.
By employing polar equations, one gains insight into how graphs behave under different angular perceptions, allowing unique visualizations and analyses. When studying a curve, observing changes in \( r \) with relation to \( \theta \) helps in grasifying potential patterns or symmetries inherent in the polar model.

Trigonometric Functions

Trigonometric functions like sine and cosine play a crucial role in the representation of curves, especially in polar equations. They describe how angles relate to ratios of side lengths in right triangles.

• Sine (\( \sin \theta \): Given a right triangle, the sine value depicts the ratio of the opposite side to the hypotenuse for angle \( \theta \). In polar equations, sine helps articulate rotational transformations.
• Cosine (\( \cos \theta \): For the same angle \( \theta \), cosine describes the ratio of the adjacent side to the hypotenuse.

These functions allow polar curves like circles or petals of flowers to be expressed compactly. Applying transformations involving these functions can significantly alter the symmetry and dimensions of curves. Additionally, understanding properties such as periodicity and even/odd nature helps predict their impacts in polar coordinates.