Polar Coordinates
Polar Coordinates
In the simulation below you can see the polar coordinate system measuring the distance between the head of the arrow and its ending point, starting at the origin, (0, 0). Notice how there are x and y vector components accompanying the vector. In polar coordinates, the components are usually calculated, whereas the radius and angle are provided. Here the angle and radius are provided along with the x and y components, however in general only the radius and angle are provided leaving the components to be determined.
Polar coordinates are a coordinate system that uses the radius r, and the angle theta on a two dimensional plane. Polar coordinates in three dimensions are known as spherical coordinates. This coordinate system will be explained in the Spherical Coordinate section. Polar coordinates are best used to measure the distance between the origin and radius of the circle as well as knowing the angle at which an object is oriented in. For example, you can measure points: away from the origin (0,0), and know which direction an object is pointing towards. It is worth noting that in Polar coordinates, the radius r, ranges from 0 to infinity, in any direction, whereas the angle theta, ranges from 0 to 2pi. The angle ranges between 0 to 2pi. The polar coordinate system differs from the cartesian coordinate system in that the distance and angle are provided, and the components are left to be calculated.
The simulation above shows the polar coordinate system used in a two dimensional plane, over the x and y axes. As you move your mouse around the simulation, you can notice the black vector, which is the radius of the vector, change in length, along with the angle theta which is provided on the side of the simulation. Polar coordinates are great for determining the distance of an object from the origin along with its orientation. For example if you move your mouse to a radius of 16.2 and an angle of 45.6 degrees, you can notice that the radius of the circle is facing the northeast direction. Knowing the radius of the vector and the angle, theta can help you determine the x and y components of the radius. These can be calculated using trigonometry, namely the sine and cosine functions.
Polar coordinates can be converted to cartesian coordinates using the following formulas:
\[ x = r \cos(\theta) \]
\[ y = r \sin(\theta) \]
\[ r = \sqrt{x^2 + y^2} \]
Proof of :
\[ r = \sqrt{x^2 + y^2} \]
\[ r = \sqrt{(r \cos(\theta))^2 + (r \sin(\theta))^2} \]
\[ r = \sqrt{(r)^2 (\cos(\theta))^2 + (r)^2 (\sin(\theta))^2} \]
\[ r = \sqrt{(r)^2 \cdot (\cos(\theta)^2 + \sin(\theta)^2)} \]
\[ r = \sqrt{(r)^2 \cdot (1)^2} \]
\[ r = \sqrt{(r)^2 \cdot 1} \]
\[ r = \sqrt{r^2} \]
\[ r = r \]
Cartesian coordinates can be converted to polar coordinates using the following formulas:
To calculate the radius, the following formulas are used:
\[ r = \frac{x}{\cos(\theta)} \]
or
\[ r = \frac{y}{\sin(\theta)} \]
To calculate the angle, the following formulas are used:
\[ \theta = \cos^{-1} \left( \frac{x}{r} \right) \]
or
\[ \theta = \sin^{-1} \left( \frac{y}{r} \right) \]
or
\[ \theta = \tan^{-1} \left( \frac{y}{x} \right) \]
Proof of :
\[ x = r \cos(\theta) \]
\[ y = r \sin(\theta) \]
\[ \frac{x}{y} = \frac{r \cos(\theta)}{r \sin(\theta)} \]
\[ x r \sin(\theta) = y r \cos(\theta) \]
\[ x \sin(\theta) = y \cos(\theta) \]
\[ \frac{\sin(\theta)}{\cos(\theta)} = \frac{y}{x} \]
\[ \tan(\theta) = \frac{y}{x} \]
\[ \tan^{-1}(\tan(\theta)) = \tan^{-1} \left( \frac{y}{x} \right) \]
\[ \theta = \tan^{-1} \left( \frac{y}{x} \right) \]
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