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?
Yes, your second answer is wrong.
polar coordinates are of the form (radius, angle) your point is backwards.
you know, as your teacher told you, that the point (-3,-3) is in quadrant III (just sketch it)
so (-3√2, π/4) is correct,
wouldn't (3√2, 5π/4) put you in the same position?
To find the polar coordinates of a point, you can use the following formulas:
r = sqrt(x^2 + y^2),
θ = atan2(y, x).
In this case, the rectangular coordinates of the point are (-3,-3). Let's calculate the polar coordinates step by step:
Step 1: Calculate the value of r.
r = sqrt((-3)^2 + (-3)^2) = sqrt(9 + 9) = sqrt(18) = 3√2.
Step 2: Calculate the value of θ.
θ = atan2(-3, -3).
The atan2 function returns an angle in radians between -π and π. To obtain an angle between 0 and 2π, you need to add 2π to the result if it is negative.
θ = atan2(-3, -3) + 2π.
Substituting the values, we get:
θ = -0.785 + 2π,
θ = 5.498.
So, the correct polar coordinates for the point (-3, -3) in quadrant 3 are r = 3√2 and θ = 5.498. Therefore, your second answer (π/4, -3√2) is incorrect because it does not fall in quadrant 3.