The coordinate (4,3) becomes (2.733,4.187) when the coordinate axes arerotated at θ counterclockwise. What is θ?

To find the angle θ, we can use the rotation matrix for a point in 2D space. The rotation matrix is given by:

[cos(θ) -sin(θ)]
[sin(θ)  cos(θ)]

When we rotate a point (x, y) using this matrix, we get the new coordinates (x', y'):

x' = xcos(θ) - ysin(θ)
y' = xsin(θ) + ycos(θ)

Given the original point (4,3) and the new point (2.733,4.187), we can set up the following system of equations:

2.733 = 4cos(θ) - 3sin(θ)
4.187 = 4sin(θ) + 3cos(θ)

This system of equations can be solved for cos(θ) and sin(θ). However, it's a bit tricky to solve directly. A more straightforward way is to use the tangent half-angle formula:

tan(θ/2) = (sin(θ)) / (1 + cos(θ))

We can solve the above system for sin(θ) and cos(θ), then substitute those into the half-angle formula to find θ.

Solving the system of equations gives:

cos(θ) = (2.733 + 4.187) / 5 = 1.384
sin(θ) = (4.187 - 2.733) / 5 = 0.291

Substituting these into the half-angle formula gives:

tan(θ/2) = 0.291 / (1 + 1.384) = 0.173

Taking the arctangent of both sides gives:

θ/2 = arctan(0.173)

Multiplying both sides by 2 gives the final answer:

θ = 2 * arctan(0.173)

This is the angle of rotation in radians. To convert to degrees, multiply by 180/π.

Please note that the above calculations are based on the assumption that the given coordinates are correct and the rotation is counterclockwise. If the rotation is clockwise, the signs of the sin(θ) terms in the equations would be reversed.