Question

Simplify the following equations and represent the results in polar coordinates. μst show your work.

a)r=√x2+y2
b)θ=arctan y/x
c)r=1/θ
d)θ=sin−1x/r

Answer 1

To simplify the equations and represent the results in polar coordinates, we convert each equation from Cartesian coordinates (x, y) to polar coordinates (r, θ). The steps for each equation are explained.

Step-by-step explanation:

To simplify the equations and represent the results in polar coordinates, we need to convert each equation from Cartesian coordinates (x, y) to polar coordinates (r, θ). Let's go through each equation:

1. a) r = sqrt(x^2 + y^2)

   This equation represents the distance from the origin to the point (x, y). It can be simplified by using the Pythagorean theorem. For example, if (x, y) = (3, 4), then r = sqrt(3^2 + 4^2) = 5.

2. b) θ = arctan(y/x)

   This equation represents the angle that line OP makes with the x-axis. To find θ, we use the arctan function. For example, if (x, y) = (3, 4), then θ = arctan(4/3) = 53.13°.

3. c) r = 1/θ

   This equation represents the inverse of the angle θ. For example, if θ = 45°, then r = 1/45° = 0.02222.

4. d) θ = sin^(-1)(x/r)

   This equation represents the angle θ that has a sine equal to x/r. For example, if (x, y) = (3, 4), then θ = sin^(-1)(3/5) = 36.87°.