I've received an assignment to investigate the polar coordinate system compared to the cartesian one.
Theres a question that asks you to investigate into how to convert polar coordinates into cartesian.
From my text book, I know in Polar coordinates, that if (d,è) is a point, where is d the distance and è is the angle, to convert into cartesian,
x = d cos è
y = d sin è
but I don't know how to prove it, or even explain it
make a diagram such as this
http://www.kwon3d.com/theory/crdsys/polar.html
To understand the conversion from polar coordinates to Cartesian coordinates, let's explain the reasoning behind it. First, let's take a look at the Cartesian coordinate system, also known as the rectangular coordinate system.
In the Cartesian coordinate system, a point is represented by its distance from the x-axis (horizontal) and the y-axis (vertical). The x-coordinate gives the horizontal position, and the y-coordinate gives the vertical position.
Now, in the polar coordinate system, a point is represented by its distance from the origin (often denoted as "d") and the angle it forms with the positive x-axis (often denoted as "θ" or "φ").
To convert from polar coordinates to Cartesian coordinates, we use trigonometry. Specifically, the sine function (sin) and the cosine function (cos).
In a right-angled triangle, the sine of an angle (θ) is defined as the ratio of the length of the side opposite the angle to the hypotenuse, and the cosine of an angle (θ) is defined as the ratio of the length of the side adjacent to the angle to the hypotenuse.
In the Cartesian coordinate system, the x-coordinate is represented by the horizontal distance (d cos θ), and the y-coordinate is represented by the vertical distance (d sin θ). Here's how we derive these equations:
1. Start with the Cartesian coordinates (x, y) and the polar coordinates (d, θ).
2. Draw a right-angled triangle with the hypotenuse from the origin to the point (x, y) and d as its length.
3. The horizontal distance from the origin to the point is given by x. This is the adjacent side of the triangle.
4. The vertical distance from the origin to the point is given by y. This is the opposite side of the triangle.
5. Using trigonometry, we know that cos θ = adjacent/hypotenuse and sin θ = opposite/hypotenuse.
6. Substituting x for the adjacent side and y for the opposite side, we get cos θ = x/d and sin θ = y/d.
7. Rearranging the equations, we get x = d cos θ and y = d sin θ.
Therefore, to convert polar coordinates (d, θ) to Cartesian coordinates (x, y), we use the equations x = d cos θ and y = d sin θ.
By using these equations, you can convert any point in polar coordinates to its corresponding Cartesian representation.