Question

How do you convert \[(5.7, - 1.2)\] into polar coordinates?

Answer 1

Hint: Here we are asked to convert the given coordinate into a polar coordinate. The polar form of the point or coordinate \[(x,y)\] is \[(r,\theta )\]. Here \[r\] is given by \[\sqrt {{x^2} + {y^2}} \] and the \[\theta \] is given by \[{\tan ^{ - 1}}\left( {\dfrac{y}{x}} \right)\] . By finding the values of \[r\] and \[\theta \] we can find the polar coordinates of \[(5.7, - 1.2)\].

Complete step-by-step solution:
It is given that \[(5.7, - 1.2)\] , this is a cartesian coordinate and we aim to find the polar coordinate of these cartesian coordinates.
We know that the polar coordinate of the cartesian coordinate \[(x,y)\] is \[(r,\theta )\] where \[r\] is the square root of the sum of squares of the coordinates \[x\& y\], \[r = \sqrt {{x^2} + {y^2}} \] and \[\theta \] is the inverse tangent of the ratio \[\dfrac{y}{x}\] that is \[\theta = {\tan ^{ - 1}}\left( {\dfrac{y}{x}} \right)\].
Here \[(x,y)\] is the given coordinate \[(5.7, - 1.2)\]. Now we will first find the value of \[r\].
From the definition of polar coordinates, we know that \[r = \sqrt {{x^2} + {y^2}} \]. Let us substitute the values of \[x\] coordinate and \[y\] coordinate.
So, \[r = \sqrt {{{(5.7)}^2} + {{( - 1.2)}^2}} \]
Let us simplify this.
          \[ = \sqrt {(5.7 \times 5.7) + (( - 1.2) \times ( - 1.2))} \]
Let’s simplify this further. We know that a square of a number can be written as multiplying that number with itself. And we also know that multiplying two negative integers will result in a positive.
          \[ = \sqrt {32.49 + 1.44} \]
       \[r = \sqrt {33.93} \]
Thus, we got the value of \[r\] now let us find the value of \[\theta \].
We know that from the definition of polar coordinates \[\theta = {\tan ^{ - 1}}\left( {\dfrac{y}{x}} \right)\]. Let us substitute the values of \[x\] coordinate and \[y\] coordinate.
Thus, \[\theta = {\tan ^{ - 1}}\left( {\dfrac{{ - 1.2}}{{5.7}}} \right)\]
Now let us multiply and divide the ratio \[\dfrac{{ - 1.2}}{{5.7}}\] by ten to remove its decimal points.
\[\theta = {\tan ^{ - 1}}\left( {\dfrac{{ - 1.2}}{{5.7}} \times \dfrac{{10}}{{10}}} \right)\]
    \[ = {\tan ^{ - 1}}\left( {\dfrac{{ - 12}}{{57}}} \right)\]
Now dividing the numerator and the denominator of the ratio by three we get
\[\theta = {\tan ^{ - 1}}\left( {\dfrac{{ - 1.2}}{{5.7}} \div \dfrac{3}{3}} \right)\]
\[\theta = {\tan ^{ - 1}}\left( {\dfrac{{ - 4}}{{19}}} \right)\]
Thus, we have found the value of \[\theta \]. Now we have both values \[r\& \theta \].
Hence, the polar coordinate of the given cartesian coordinate \[(5.7, - 1.2)\] is \[\left( {\sqrt {33.93} ,{{\tan }^{ - 1}}\left( {\dfrac{{ - 4}}{{19}}} \right)} \right)\].

Note: When there is a rational number the equivalent rational number of that can be found by multiplying or dividing the numerator and denominator of that rational number by any number. Here we have multiplied the ratio \[\dfrac{{ - 1.2}}{{5.7}}\] by \[10\] to get an equivalent rational number thus it doesn’t change the original ratio.