Question

What are the polar coordinates of the point with \[( - 3,3)\] rectangle coordinate?
(A) \[(3,135)\]
(B) \[( - 3,135)\]
(C) \[(3\sqrt 2 ,45)\]
(D) \[(3\sqrt 2 ,135)\]
(E) \[( - 3\sqrt 2 ,45)\]

Answer 1

Hint: A polar coordinate system is a two dimensional coordinate system in which each point on a plane is determined by a distance from the reference point and an angle from a reference direction. Polar coordinates is a pair of coordinates locating the position of a point in a plane, first being the distance from a fixed point on a line and the second is the angle made by this line with a fixed line. In Cartesian coordinates there is exactly one set of coordinates for a given point. Cartesian coordinates are written as \[(x,y)\] whereas polar coordinates are written as \[(r,\theta )\] .
Here \[x,y\] are the respective values of abscissa and ordinate .
So ,\[r = \sqrt {{x^2} + {y^2}} \] ,
\[\tan \theta = \dfrac{y}{x}\]
Therefore, \[\theta = {\tan ^{ - 1}}\dfrac{y}{x}\]

Complete step-by-step answer:
We know Cartesian coordinates are represented by \[(x,y)\] where \[x\] is the abscissa and \[y\] is the ordinate.
Here, in this question, the given coordinates are \[( - 3,3)\] .
So the abscissa ,\[x = - 3\].
And the ordinate ,\[y = 3\].
Now we have to convert it to polar coordinates \[(r,\theta )\] .
Where \[r\] is the distance from a fixed point of a line and \[\theta \] is the angle made by this line with a fixed line.
We know ,\[r = \sqrt {{x^2} + {y^2}} \] .
Putting the value of \[x\] and \[y\] in the formula of \[r\] , we get,
So, \[r = \sqrt {{{( - 3)}^2} + {{(3)}^2}} \] .
\[ \Rightarrow r = \sqrt {9 + 9} \]
\[ \Rightarrow r = 3\sqrt 2 \]
We know ,\[\tan \theta = \dfrac{y}{x}\]
Therefore \[\theta = {\tan ^{ - 1}}\dfrac{y}{x}\]
Putting the value of \[x\] and \[y\] in the formula of \[\theta \] , we get.
So, \[\theta = {\tan ^{ - 1}}\dfrac{{ - 3}}{3}\]
\[ \Rightarrow \theta = {\tan ^{ - 1}}( - 1)\]
\[ \Rightarrow \theta = \dfrac{{3\pi }}{4} = \dfrac{{7\pi }}{4}\] .
Since, \[\tan \dfrac{{3\pi }}{4} = - 1\] and \[\tan \dfrac{{7\pi }}{4} = - 1\] .
As \[( - 3,3)\] lies in 2nd quadrant so, \[\theta = \dfrac{{3\pi }}{4}\] ,Since in 2nd quadrant \[x\] is negative and \[y\] is positive .
Therefore, \[r = 3\sqrt 2 \]
\[\theta = \dfrac{{3\pi }}{4}\]
\[ \Rightarrow \theta = \dfrac{{3 \times 180}}{4}\]
hence, \[\theta = 135^\circ \]
So ,The polar coordinates of the point with \[( - 3,3)\] rectangle coordinates are \[(3\sqrt 2 ,135^\circ )\] .
So, the correct answer is “Option D”.

Note: We should be careful in calculating the value of \[\theta \] as it depends on the quadrant. Avoid mistakes regarding the sign of the value of abscissa and ordinate. In the first quadrant , the value of all trigonometric identities is always positive . Whereas in the second quadrant the value of sin and cosec is always positive , in the third quadrant the value of tan and cot is always positive and in the fourth quadrant the value of cos and sec is always positive.