Question

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

Short answer

Question: Determine the symmetries (if any) of the curve given by the polar equation $r=4-\sin (\theta /2)$ without using a graphing utility. Answer: The curve given by the polar equation $r=4-\sin (\theta /2)$ does not possess any symmetries.

Step-by-step solution

Test for Symmetry about the x-axis

To check for symmetry about the x-axis, we will replace $\theta$ with $-\theta$ and check if the polar equation remains the same. $r=4-\sin ((-\theta )/2)$ $r=4-\sin (-\theta /2)$ Since $\sin (-x)=-\sin (x)$, $r=4+\sin (\theta /2)$ The original equation is not equal to this new equation, so there is no symmetry about the x-axis.

Test for Symmetry about the y-axis

To check for symmetry about the y-axis, we will replace $\theta$ with $(\pi -\theta )$ and check if the polar equation remains the same. $r=4-\sin ((\pi -\theta )/2)$ Using the sin subtraction formula, $\sin (a-b)=\sin a\cos b-\cos a\sin b$, we have $r=4-\sin (\pi /2-\theta /2)$ $r=4-(\sin (\pi /2)\cos (\theta /2)-\cos (\pi /2)\sin (\theta /2))$ Since, $\sin (\pi /2)=1$ and $\cos (\pi /2)=0$, we have $r=4-\cos (\theta /2)$ The original equation is not equal to this new equation, so there is no symmetry about the y-axis.

Test for Symmetry about the Origin

To check for symmetry about the origin, we will replace $\theta$ with $(\theta +\pi )$ and check if the polar equation remains the same. $r=4-\sin ((\theta +\pi )/2)$ $r=4-\sin (\theta /2+\pi /2)$ Using the sin addition formula, $\sin (a+b)=\sin a\cos b+\cos a\sin b$, we have $r=4-(\cos (\theta /2)\sin (\pi /2)+\sin (\theta /2)\cos (\pi /2))$ Since, $\sin (\pi /2)=1$ and $\cos (\pi /2)=0$, we have $r=4-\cos (\theta /2)$ The original equation is not equal to this new equation, so there is no symmetry about the origin. In conclusion, the curve given by the polar equation $r=4-\sin (\theta /2)$ does not possess any symmetries.

Key concepts

Symmetry

In the context of polar coordinates, symmetry is a property that can significantly simplify the process of graphing equations. When a curve is symmetrical about a particular axis or point, it means that a portion of the graph can be mirrored across that axis or point to produce the remaining part of the graph. This concept is invaluable when dealing with complex curves, as identifying symmetry helps in sketching accurate graphs without needing detailed calculations across the entire range of angles. There are three primary types of symmetry to test in polar equations:

• Symmetry about the x-axis: Replace $\theta$ with $-\theta$ in the equation. If the resulting equation is identical to the original, the curve is symmetrical about the x-axis.
• Symmetry about the y-axis: Replace $\theta$ with $\pi -\theta$. If there's no change in the equation, the curve shows symmetry about the y-axis.
• Symmetry about the origin: Use $\theta +\pi$. Symmetry about the origin means the equation remains unchanged under this transformation.

For the given equation $r=4-\sin (\theta /2)$, testing all these symmetries reveals that it does not possess any of them.

Trigonometric Identities

Trigonometric identities are crucial tools in simplifying expressions and solving equations in polar coordinates. These identities provide relationships between trigonometric functions that allow us to transform and manipulate equations, making complexities more manageable. Key identities utilized include:

• Sine and cosine angle sum and difference: These formulas, like $\sin (a\pm b)$ and $\cos (a\pm b)$, allow us to break down complex arguments into simpler components.
• Even and odd properties: The function $\sin (-\theta )=-\sin (\theta )$ is an odd function, affecting symmetry testing, while $\cos (-\theta )=\cos (\theta )$ is even.

In the given exercise, these identities help to evaluate the symmetries by transforming the original polar equation when replacing $\theta$ to test different symmetries.

Polar Equations

Polar equations, like $r=4-\sin (\theta /2)$, offer a unique perspective on representing curves in a coordinate system where each point is determined by a distance from a central point and an angle from a reference direction. Unlike Cartesian coordinates where you have $(x,y)$ pairs, polar coordinates use $(r,\theta )$ where $r$ is the radial distance and $\theta$ is the angle.Polar equations are especially useful when dealing with curves that exhibit rotational symmetry or when you are dealing with circular or spiral patterns which would be cumbersome in Cartesian formats. These equations can be challenging due to the need to manage trigonometric functions, but they also provide a more natural way to model certain kinds of phenomena. When exploring polar equations, always consider:

• The basic form: Simple equations can exhibit complex curves.
• Test symmetries: This simplifies the process of sketching these curves.

Sin Addition Formula

The sin addition formula is a mathematical identity that provides a method to expand expressions involving the sine function with multiple angles. It is written as $\sin (a+b)=\sin a\cos b+\cos a\sin b$. This identity is essential for simplifying complex trigonometric expressions and is used extensively in calculus and precalculus.In the context of the exercise, the sin addition formula was implemented to determine the symmetry of polar equations by rewriting angles sum as products of sines and cosines. Here's how it works:

• When you have a sum such as $\theta +\pi$, using the formula can break down this angle into parts that are easier to analyze in terms of radial equations.
• This helps check if the resulting expression maintains its original form, indicating symmetry.

The sin addition formula is fundamental in both confirming and refuting symmetry when analyzing polar equations.