The cartesian coordinates of a point are given. (a) (4, −4)(i) find polar coordinates (r, θ) of the point, where r > 0 and 0 ≤ θ < 2π.(ii) find polar coordinates (r, θ) of the point, where r < 0 and 0 ≤ θ < 2π.(b) (−1, sqrt() find polar coordinates (r, θ) of the point, where r > 0 and 0 ≤ θ < 2π.(ii) find polar coordinates (r, θ) of the point, where r < 0 and 0 ≤ θ < 2π.

i)

ii)

Thinking Process:

This can either be solved using complex variables or simply trigonometry (they are all the same in a way)

But we'll simply go with trigonometry and good old Pythagorean theorem.

Solution:

For Cartesian coordinates are:

To covert them into

We can use:

These are related since:

So, let's start solving:

Part a) (4,-4)

For r:

i) for we'll go with

ii) for we'll go with

For theta:

One thing to keep in mind is that the point (4,-4) lies in the 4th quadrant of the xy-plane, and the range given for is . So we have to add to our answer (we have gone around the circle and stopped at the same place as radians)

radians

radians

The Answer for part(a) is

Part b) (-1, sqrt(3))

For r:

i) for

ii) for

For theta:

this not in range of theta: .

and the point lies in the 2nd quadrant, so we can add radians to our answer

in range:

The Answer for part(b) is

-7

step-by-step explanation:

x^2 + 18x + 74 = (x + 9)^2 - 81 + 74

= (x + 9)^2 - 7.

so q = -7.