The rectangular coordinates of a point are given. Find two sets of polar coordinates for the point for 0 is less than or equal to Q is less than 2pi.
(-3,-3)
My answers were (-3sqrt2, pi/4) and (pi/4, -3sqrt2).
My teacher said that my answers have to be in quadrant 3. Is my second answer wrong?
To find the polar coordinates of a point given its rectangular coordinates, you can use the following formulas:
r = sqrt(x^2 + y^2)
θ = arctan(y / x)
Let's apply these formulas to the given point (-3, -3):
1. Calculate the value of r:
r = sqrt((-3)^2 + (-3)^2) = sqrt(18) = 3√2
2. Calculate the value of θ:
θ = arctan((-3) / (-3)) = arctan(1) = π/4
Now, let's analyze your answers:
1. (-3√2, π/4): This represents a point in the second quadrant, not in the third quadrant. The value of r seems to be correct, but the angle θ is incorrect for the third quadrant.
2. (π/4, -3√2): This represents a point in the fourth quadrant, not in the third quadrant. Both the value of r and the angle θ are incorrect for the third quadrant.
Since your teacher instructed you to find polar coordinates in the third quadrant, you need to adjust the angle θ accordingly. In the third quadrant, the angle is between π and (3π/2).
The correct polar coordinates for the point (-3, -3) in the third quadrant are:
(3√2, 5π/4) and (3√2, -3π/4)
These coordinates represent the magnitude (r) and angle (θ) in polar form, with r being positive in both cases.