what is (sqrt 2, -pi/4) in rectangular coordinates?
what is (-sqrt 6, -sqrt 2) in polar coordinates?
To convert a point from polar coordinates to rectangular coordinates, we use the following formulas:
x = r * cos(theta)
y = r * sin(theta)
where r represents the radial distance from the origin to the point, and theta represents the angle in radians measured from the positive x-axis to the line segment connecting the origin to the point.
Conversely, to convert a point from rectangular coordinates to polar coordinates, we use the following formulas:
r = sqrt(x^2 + y^2)
theta = atan2(y, x)
where x and y are the Cartesian coordinates of the point.
Now let's solve the first problem:
1. (sqrt(2), -pi/4) in rectangular coordinates:
Using the given polar coordinates, we have:
r = sqrt(2) (square root of 2)
theta = -pi/4 (negative pi/4)
Using the conversion formulas for rectangular coordinates:
x = r * cos(theta) = sqrt(2) * cos(-pi/4) = sqrt(2) * (1/sqrt(2)) = 1
y = r * sin(theta) = sqrt(2) * sin(-pi/4) = sqrt(2) * (-1/sqrt(2)) = -1
So, (sqrt(2), -pi/4) in rectangular coordinates is (1, -1).
Now let's solve the second problem:
2. (-sqrt(6), -sqrt(2)) in polar coordinates:
Using the given rectangular coordinates, we have:
x = -sqrt(6) (negative square root of 6)
y = -sqrt(2) (negative square root of 2)
Using the conversion formulas for polar coordinates:
r = sqrt(x^2 + y^2) = sqrt((-sqrt(6))^2 + (-sqrt(2))^2) = sqrt(6 + 2) = sqrt(8) = 2 * sqrt(2)
theta = atan2(y, x) = atan2(-sqrt(2), -sqrt(6))
Note: The atan2 function gives the angle in the correct quadrant.
Using a calculator or software:
theta = atan2(-sqrt(2), -sqrt(6)) ≈ -0.588 (approximately)
So, (-sqrt(6), -sqrt(2)) in polar coordinates is (2 * sqrt(2), -0.588).