Question

express each of the following in standard cartesian form, i.e., x iy (its ok to use polar form as an intermediate step)

Answer 1

To convert polar coordinates (r, θ) to Cartesian coordinates, use the equations x = r × cos(θ) and y = r × sin(θ).

Step-by-step explanation:

To express a point or vector given in polar coordinates (r, θ) into Cartesian coordinates, we can apply the relationships that define cosine and sine functions:
x = r × cos(θ)
y = r × sin(θ)

These equations allow us to convert from polar to Cartesian form, expressing the coordinates as x + iy, where i is the unit vector along the x-axis. This is useful in a variety of mathematical contexts, particularly when describing rotations or when objects are moving along curves. For example, if we have a vector with polar coordinates (5, π/4), the Cartesian form would be approximately 3.54 + 3.54i.