The rectangular coordinates of the point P are (-1,sqrt 3). If P is reflected witg respect to the x-axis, then the polar coordinates of its image are¡??
(-1 , √3) ----> (-1 , -√3) after reflection in the x-axis
new point is in III,
tanØ = -√3/-1 = 240° in III
magnitude = √(1^2 + √3^2) = √4 = 2
point in polar coordinates:
(2cos240°, 2sin240°) or (2cos 4π/3, 2sin 4π/3)
To find the polar coordinates of the reflected point, we need to first determine the corresponding Cartesian coordinates of the reflected point.
If P is reflected with respect to the x-axis, the y-coordinate will change its sign while the x-coordinate remains the same. In this case, the y-coordinate of P is sqrt(3), so the reflected point will have a y-coordinate of -sqrt(3), and the x-coordinate will still be -1.
Thus, the Cartesian coordinates of the reflected point Q would be (-1, -sqrt(3)).
To find the polar coordinates of Q, we can use the formulas:
r = sqrt(x^2 + y^2)
theta = arctan(y / x)
Substituting the values, we have:
r = sqrt((-1)^2 + (-sqrt(3))^2)
= sqrt(1 + 3)
= sqrt(4)
= 2
theta = arctan((-sqrt(3)) / -1)
= arctan(sqrt(3))
= 60 degrees
Therefore, the polar coordinates of the reflected point are (r, theta) = (2, 60 degrees).