What is the double angle formula in polar coordinates?

The double angle formula in polar coordinates is used to find the coordinates of a point that is twice the angle of another point.

To derive the double angle formula in polar coordinates, we start with the equation for a point in polar coordinates:

r = sqrt(x^2 + y^2)
theta = arctan(y/x)

To find the coordinates of a point that is twice the angle of another point, we can use the identity:

cos(2theta) = cos^2(theta) - sin^2(theta)
sin(2theta) = 2sin(theta)cos(theta)

Using this identity, we can express the coordinates of the point that is twice the angle of another point in terms of the coordinates of the original point:

r' = sqrt(x^2 + y^2)
theta' = 2theta

x' = r'cos(theta') = r'cos(2theta) = r(cos^2(theta) - sin^2(theta))
y' = r'sin(theta') = r'sin(2theta) = 2rsin(theta)cos(theta)

Therefore, the double angle formula in polar coordinates is:

x' = r(cos^2(theta) - sin^2(theta))
y' = 2rsin(theta)cos(theta)

This formula can be used to find the coordinates of a point that is twice the angle of another point, which can be useful in a variety of applications in mathematics and physics.