Question

For the following exercises, sketch the polar curve and determine what type of symmetry exists, if any. $$ r=5 \cos (5 \theta) $$

Short answer

The curve has 5 petals and is symmetric about the polar axis.

Step-by-step solution

Convert Polar Equation into Cartesian Coordinates

Before sketching the polar curve, it's helpful to convert the given polar equation into Cartesian coordinates to better understand its shape. The polar equation is \( r = 5 \cos(5\theta) \). To convert to Cartesian, use the formulas:- \( x = r\cos\theta \)- \( y = r\sin\theta \)Substituting the polar form, we have:\[ x = 5\cos(5\theta)\cos\theta = \frac{5}{2}(\cos(6\theta) + \cos(4\theta)) \]It's complex to sketch directly from this, so proceed to sketch with polar coordinates.

Identify and Sketch the Polar Curve

The curve defined by the equation \( r = 5 \cos(5\theta) \) is known as a rose curve. The form \( r = a \cos(n\theta) \) results in a rose with petals, where:- If \( n \) is even, the rose has \( 2n \) petals.- If \( n \) is odd, the rose has \( n \) petals.Here, \( n = 5 \) (an odd number), so there are 5 petals.To sketch, plot points for various values of \( \theta \) within \( [0, 2\pi] \), and connect to form the petal shape.

Determine Symmetry

The symmetry of polar curves can be identified around the pole, line \( \theta = \frac{\pi}{2} \), or the polar axis. For \( r = 5 \cos(5\theta) \):1. **Polar Axis Symmetry**: Replace \( \theta \) with \(-\theta\), giving the same function \( r = 5\cos(5(-\theta)) = 5\cos(5\theta) \). This indicates symmetry about the polar axis.Thus, the curve is symmetric with respect to the polar axis.

Key concepts

Polar Coordinates

Polar coordinates offer a unique way of describing the position of a point in a plane using the distance from a reference point and an angle from a reference direction. In polar coordinates, a point is defined by \(r, \theta\), where \(r\) is the radius, or distance from the origin, and \(\theta\) is the angle measured in radians from the positive x-axis.

• The origin is called the pole.
• The positive x-axis is generally referred to as the polar axis.
• Angles are typically measured in radians, but can be in degrees.

This system is particularly useful for sketching and analyzing curves that are naturally circular or exhibit rotational symmetry, such as spirals and rose curves.

By understanding how polar coordinates work, you can easily plot complex shapes and curves that might be difficult to represent using Cartesian coordinates alone.

Cartesian Coordinates

While polar coordinates use a radius and angle, Cartesian coordinates rely on an x and y coordinate system to locate a point in the plane. The two systems are related, and converting between them can help to understand a curve's shape more deeply. The conversion formulas are:

• For x: \(x = r \cos \theta\)
• For y: \(y = r \sin \theta\)

These equations are derived from the relationships in a right triangle, using trigonometric functions. Converting a polar equation into Cartesian coordinates can reveal different aspects of the curve's nature. However, as seen in the exercise, not every polar curve can be easily represented in Cartesian form.

In such cases, sticking to the polar form allows for clearer visual representation and interpretation of the curve.

Symmetry

Symmetry in polar curves is about identifying if a shape repeats itself about a line or a point. Symmetric curves are often easier to analyze because understanding one part can grant insights into other parts.

• **Polar Axis Symmetry**: A curve is symmetric with respect to the polar axis if replacing \(\theta\) with \(-\theta\) in the equation results in the same equation.
• **Line \(\theta = \frac{\pi}{2}\) Symmetry**: A curve has this symmetry if replacing \(r\) with \(-r\) results in the same equation when \(\theta\) is replaced by \(\pi - \theta\).
• **Pole Symmetry**: A curve is pole symmetric if replacing \(r\) with \(-r\) in the equation results in the same equation.

For the curve \(r = 5 \cos (5\theta)\), checking these conditions helps establish that it is symmetric around the polar axis, simplifying the process of sketching the curve.

Rose Curve

Rose curves, such as the one defined by \(r = a \cos(n\theta)\) or \(r = a \sin(n\theta)\), create beautiful shapes with repeating petals. These petals' number and distribution depend directly on the value of \(n\), a defining parameter of the curve.

• If \(n\) is odd, the curve will have \(n\) petals.
• If \(n\) is even, the curve will have \(2n\) petals.
• The parameter \(a\) affects the size of the petals.

This periodic repetition is what gives rose curves their pleasing, floral appearance, making them fascinating to study in mathematics. Since the petal count directly relates to \(n\), visualizing and drawing these curves becomes an engaging process by plotting a few values of \(\theta\) and connecting the subsequent points.