using basic trigonometry derive the following equations, which are used to convert cartesian to polar coordinates and vice versa. why is the cos(phi)in the x term?

Question

using basic trigonometry derive the following equations, which are used to convert cartesian to polar coordinates and vice versa. why is the cos(phi)in the x term?

x = R cos (phi) sin (lamda)
y = R sin (phi)

they are used in orthogonal projection of a spherical surface to a flat surface in the (x y) plane.

where (phi)=latitude
(lamda)=longitude

Answer 1

wikipedia has a nice article explaining this projection. Your example uses an origin of (0,0) for the projection, reducing the complexity of the equations considerably.

Answer 2

Graph the oriented angle in standard position.

7π
4

Answer 3

To derive the equations used to convert Cartesian (x, y) coordinates to polar coordinates (R, φ), and vice versa, we can use basic trigonometry and the concept of orthogonal projection.

First, let's convert polar coordinates to Cartesian coordinates:

Given:
R = radius
φ = latitude
λ = longitude

1. Express x in terms of R, φ, and λ:
Since x represents the horizontal distance, we can use the cosine function.
x = R * cos(φ) * sin(λ)

The reason why we use cos(φ) in the x term is because cos(φ) represents the horizontal projection of the radial distance (R) onto the x-axis. It accounts for the effect of latitude on the x-coordinate.

2. Express y in terms of R, φ, and λ:
Since y represents the vertical distance, we can use the sine function.
y = R * sin(φ)

Here, the latitude φ directly determines the vertical position (y) of the point, so it is used as is in the equation.

Now, let's convert Cartesian coordinates to polar coordinates:

Given:
x = horizontal distance
y = vertical distance

1. Find R (radius):
Using the Pythagorean theorem, we can find the magnitude (R) of the vector from the origin to the point (x, y).
R = √(x^2 + y^2)

This equation calculates the distance (R) between the origin and the point, giving the magnitude or radial coordinate.

2. Find φ (latitude):
We need to determine the angle (φ) between the positive x-axis and the line connecting the origin and the point (x, y).
φ = atan2(y, x)

Here, we use the inverse tangent function (atan2) to calculate the angle, which takes into account the signs of x and y to determine the correct quadrant in the Cartesian coordinate system.

Therefore, the equations x = R * cos(φ) * sin(λ) and y = R * sin(φ) are derived using basic trigonometry and the concept of orthogonal projection.