The formulas to convert Cartesian coordinates to polar coordinates are:
\[ r = \sqrt{x^2 + y^2} \]
\[ \theta = \arctan \left( \frac{y}{x} \right) \]
where \( r \) is the radius, \( x \) and \( y \) are the coordinate points, and \( \theta \) is the angle.
Polar coordinates are a system used to represent points in a two-dimensional plane. Unlike the commonly used Cartesian coordinates (x, y), polar coordinates express a point’s position using two values: the distance from a fixed point called the origin, and the angle between a reference direction, usually the positive x-axis, and a line connecting the origin to the point.
In polar coordinates, a point is represented by (r, θ), where \( r \) is the distance from the origin to the point and \( \theta \) is the angle measured in radians. Radians are a unit of angular measure that offers a more natural and mathematically convenient way to work with angles compared to degrees.
The importance of polar coordinates lies in their ability to simplify certain mathematical calculations and describe certain phenomena more elegantly. They are particularly useful in fields such as physics, engineering, and mathematics, where circular or rotational symmetry is present. Polar coordinates make it easier to analyze and understand such symmetrical patterns.
Let's assume we have the following values:
Step 1: Calculate the radius (\( r \)):
\[ r = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]
Step 2: Calculate the angle (\( \theta \)) in radians:
\[ \theta = \arctan \left( \frac{4}{3} \right) \approx 0.93 \text{ radians} \]
Therefore, the polar coordinates are \( (r, \theta) = (5, 0.93 \text{ radians}) \).
