The cartesian coordinates of a point are given. (a) (2, −2) (i) find polar coordinates (r, θ) of the point, where r > 0 and 0
≤ θ < 2π. (r, θ) = (ii) find polar coordinates (r, θ) of the point, where r < 0 and 0 ≤ θ < 2π. (r, θ) = (b) (−1, 3 ) (i) find polar coordinates (r, θ) of the point, where r > 0 and 0 ≤ θ < 2π. (r, θ) = (ii) find polar coordinates (r, θ) of the point, where r < 0 and 0 ≤ θ < 2π. (r, θ) =
We know that the relationship between the 2-dimensional polar and Cartesian coordinates is
r = √(x² + y²)
Θ = tan⁻¹ (y/x) so
Part a) <span>(2, −2)---------> this point belong to the IV quadrant </span>r = √(x² + y²)------ r = √(2² + (-2)²)-----> r=√8 Θ = tan⁻¹ (y/x)---- Θ = tan⁻¹ (2/2)----> 45° remember that the point belong to the IV quadrant so Θ=360-45-----> Θ=315°
the answer part A) is (r,Θ)=(√8,315°)
Part b) (-1, 3)---------> this point belong to the II quadrant r = √(x² + y²)------ r = √(-1² + (3)²)-----> r=√10 Θ = tan⁻¹ (y/x)---- Θ = tan⁻¹ (3/1)----> 71.57° remember that the point belong to the II quadrant so Θ=180-71.57-----> Θ=108.43°
Using the table, you are looking at a ratio of 9 Dimes to 6 nickels or 9:6 which can be simplified by dividing each by three (a number they are both divisible by without going into decimals) to get a ratio of 3 dimes for every 2 nickels she has, or 3:2
