Answer:
The polar coordinate of [tex]P(x,y) = (-3.50\,m,-2.50\,m)[/tex] is [tex]P (r,\theta) = (4.301\,m, 215.538^{\circ})[/tex].
Explanation:
Given a point in rectangular form, that is [tex]P(x,y) = (x,y)[/tex], its polar form is defined by:
[tex]P(x,y) = (r,\theta)[/tex] (1)
Where:
[tex]r[/tex] - Norm, measured in meters.
[tex]\theta[/tex] - Direction, measured in sexagesimal degrees.
The norm of the point is determined by Pythagorean Theorem:
[tex]r = \sqrt{x^{2}+y^{2}}[/tex] (2)
And direction is calculated by following trigonometric relation:
[tex]\theta = \tan^{-1} \frac{y}{x}[/tex] (3)
If we know that [tex]x = -3.50\,m[/tex] and [tex]y = -2.50\,m[/tex], then the components of coordinates in polar form is:
[tex]r = \sqrt{(-3.50\,m)^{2}+(-2.50\,m)^{2}}[/tex]
[tex]r \approx 4.301\,m[/tex]
Since [tex]x < 0\,m[/tex] and [tex]y < 0\,m[/tex], direction is located at 3rd Quadrant. Given that tangent function has a period of 180º, we find direction by using this formula:
[tex]\theta = 180^{\circ}+\tan^{-1} \left(\frac{-2.50\,m}{-3.50\,m} \right)[/tex]
[tex]\theta \approx 215.538^{\circ}[/tex]
The polar coordinate of [tex]P(x,y) = (-3.50\,m,-2.50\,m)[/tex] is [tex]P (r,\theta) = (4.301\,m, 215.538^{\circ})[/tex].
