Problem 85 - Essential Calculus

Question

Verify that if the curve whose polar equation is \(r=f(\theta)\) is rotated about the pole through an angle \(\phi,\) then an equation for the rotated curve is \(r=f(\theta-\phi)\)

Step-by-Step Solution

Answer

Therefore, we can conclude that a rotation of the curve \(r = f(\theta)\) about its pole by an angle \(\phi\) leads to a new curve described by the equation \(r = f(\theta - \phi)\).

Step 1: Write out the problem statement

We need to prove that if the curve with polar equation is \(r = f(\theta)\), is rotated about the pole through an angle \(\phi\), then an equation for the rotated curve is \(r = f(\theta - \phi)\).

Step 2: Expressing rotation using old polar coordinates

The polar coordinates of a point (r, \theta) on the original curve will become (r, \theta + \phi) once the curve is rotated counter-clockwise by an angle \phi.

Step 3: Create new polar equation for rotated curve

So, the polar coordinates of a point on the new (rotated) curve is (r, \theta + \phi). In order to express r as a function of \theta, we substitute \(\theta + \phi\) with \(\theta\) and we get \(r = f(\theta)\). This means the equation for the rotated curve is \(r = f(\theta - \phi)\).

Key Concepts

Curve Rotation

When we talk about curve rotation in polar coordinates, we're referring to the action of rotating a curve around the central point or pole. This is analogous to spinning a shape around its center. In polar coordinates, a point is represented as \((r, \theta)\), where \(r\) is the distance from the origin, and \(\theta\) is the angle from a reference direction—usually the positive x-axis.

When a curve defined by a polar equation \(r = f(\theta)\) is rotated through an angle \(\phi\), it involves changing the angle of each point by \(\phi\). Essentially, each point on the curve with angle \(\theta\) gets transformed to a new angle \(\theta + \phi\), moving the curve around the pole.

• Imagine a clock: rotating a point from 3 o'clock to 6 o'clock is similar to changing its angle by 90 degrees.
• Every point maintains its distance \(r\) from the center, ensuring the shape and size of the curve are unchanged, just rotated.

To determine the new equation, one replaces \(\theta\) with \(\theta + \phi\) to account for the rotation, but in order to express it in terms of the original \(\theta\), we adjust this substitution to \(\theta - \phi\) to reflect the counter-clockwise convention.

Polar Equations

Polar equations provide a way to describe curves using polar coordinates. In contrast to Cartesian equations that use \(x\) and \(y\) coordinates, polar equations express relationships based on radius \(r\) from the origin and angle \(\theta\).

These equations offer a natural and often simpler approach to defining curves that are round or involve rotational symmetry. For example, a simple circle centered at the origin can be expressed by the equation \(r = a\), where \(a\) is the radius; it's a constant value representing all points equidistant from the pole.

• An example of a more complex polar function is \(r = 1 + \sin(\theta)\), which depicts a limaçon shape.
• Many curves, such as spirals and roses, have elegant representations in polar coordinates that can be cumbersome in Cartesian coordinates.
Polar equations are particularly useful when rotation is involved, as they allow seamless transformations through angle manipulations.

Transformation of Coordinates

Transforming coordinates is about changing the framework or perspective from which we view the points on a curve. In polar coordinates, this often involves changing how we express the angle \(\theta\) or the radius \(r\).

When a transformation involves rotation, like in our original problem, it means shifting the angle \(\theta\) to \(\theta + \phi\). However, to keep the equation recognizable in terms of the original curve's expression, it results in modifying the polar equation to \(r = f(\theta - \phi)\).

• This transformation preserves the radial distances \(r\), keeping the curve's shape intact while only altering its position.
• Coordinate transformation helps in visualizing the curve from different angles or translates how the curve would look if spun around its origin.

By grasping the transformation process, one can better understand how different equations correspond to different orientations and perspectives of the same curve.