91Ó°ÊÓ

Polar coordinates are a two-dimensional coordinate system where each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point is known as the pole, typically corresponding to the origin of the Cartesian coordinate system, and the reference direction originates from the pole and is usually the positive x-axis.

Each point is represented as \( (r, \theta) \) where \( r \) is the radial distance from the pole, and \( \theta \) is the polar angle measured in radians from the reference direction. Polar coordinates are particularly useful for dealing with problems involving circular and spiral shapes, where using Cartesian coordinates can be cumbersome.

Polar equations are mathematical expressions that provide a relationship between the radius \( r \) and the angle \( \theta \) in polar coordinates. These equations describe curves on the polar plane. A simple example is the equation of a circle \( r = a \) where \( a \) is the radius of the circle.

The lima\c{c}on that is given in the exercise, \( r=2-4\sin(\theta) \), is a more complex shape that depends on both the angle and the sine function. This equation showcases the behavior of the radius changing in accordance with the angle \( \theta \) to form a loop or dimple, characteristic of a lima\c{c}on, in polar coordinates.

Integration in polar coordinates is a technique used to calculate areas and other properties of shapes that are easily described by polar equations. To find the area \( A \) enclosed by a polar curve \( r(\theta) \) between two angles \( \alpha \) and \( \beta \) we use the integral:

\( A = \frac{1}{2}\int_{\alpha}^{\beta} r(\theta)^2 d\theta \).

The factor of \( \frac{1}{2} \) comes from the derivation of the area element in polar coordinates, \( \frac{1}{2}r^2d\theta \) which represents a tiny wedge-shaped section of the area. Summing up (integrating) all such wedges between \( \alpha \) and \( \beta \) gives us the total area. In our exercise, this approach was used to determine the area bounded by the lima\c{c}on curve and the origin, through setting up and evaluating the definite integral.

Double-angle identities are formulas in trigonometry that express trigonometric functions of double angles in terms of trigonometric functions of single angles. One of the most widely used double-angle identities is for the sine squared function:

\( \sin^2(\theta) = \frac{1}{2}(1 - \cos(2\theta)) \).

These identities are particularly useful when dealing with integrals involving trigonometric functions, such as while integrating in polar coordinates to find areas. In the case of our exercise, the double-angle identity allowed the transformation of the \( \sin^2(\theta) \) term into an expression involving \( \cos(2\theta) \) which can be easily integrated to find the area under the lima\c{c}on curve.