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Question: How do I convert polar coordinates \(\left( 6,60{}^\circ \right)\) to rectangular coordinates?...

How do I convert polar coordinates (6,60∘)\left( 6,60{}^\circ \right) to rectangular coordinates?

Explanation

Solution

Like the rectangular coordinates, the polar coordinates are also a method for determining the position of a point in a plane. The rectangular coordinates are represented by the ordered pair (x,y)\left( x,y \right) whereas the polar coordinates are represented by the ordered pair (r,θ).\left( r,\theta \right). These different coordinates are related by the relations, x=rcos⁡θx=r\cos \theta and y=rsin⁡θ.y=r\sin \theta .

Complete step by step answer:
Let us consider the given data.
We are given with the polar coordinates (6,60∘).\left( 6,60{}^\circ \right).
We know that like the rectangular coordinates, the polar coordinates are also a method of determining the position of a point in a plane.
And the general representation of the polar coordinate is (r,θ).\left( r,\theta \right).
In this ordered pair, rr is the distance of the point from the origin and θ\theta is the angle made by the line joining the point and the origin with the x−x-axis.
Now, we can find the corresponding rectangular coordinates, represented by the ordered pair (x,y)\left( x,y \right) using the relations x=rcos⁡θx=r\cos \theta and y=rsin⁡θ.y=r\sin \theta .
Here, in our case, r=6r=6 and θ=60∘.\theta =60{}^\circ .
Let us substitute these values in the relations written above to get the corresponding rectangular coordinates.
So, first let us find the coordinate xx using the relation x=rcos⁡θ.x=r\cos \theta .
We will get x=6cos⁡60∘.x=6\cos 60{}^\circ .
We know that cos⁡60∘=12.\cos 60{}^\circ =\dfrac{1}{2}.
So, x=6×12=3.x=6\times \dfrac{1}{2}=3.
Similarly, let us find the coordinate yy using the relation y=rsin⁡θ.y=r\sin \theta .
We will get y=6sin⁡60∘.y=6\sin 60{}^\circ .
We have learnt that sin⁡60∘=32.\sin 60{}^\circ =\dfrac{\sqrt{3}}{2}.
This will give us the value as y=6×32=33.y=6\times \dfrac{\sqrt{3}}{2}=3\sqrt{3}.

Hence the rectangular coordinates corresponding to the polar coordinates (6,60∘)\left( 6,60{}^\circ \right) are (3,33).\left( 3,3\sqrt{3} \right).

Note: We know that x=rcos⁡θx=r\cos \theta and y=rsin⁡θ.y=r\sin \theta . Now, we will get x2+y2=r2.{{x}^{2}}+{{y}^{2}}={{r}^{2}}. This can be proved as r2cos⁡2θ+r2sin⁡2θ=r2(cos⁡2θ+sin⁡2θ)=r2×1=r2.{{r}^{2}}{{\cos }^{2}}\theta +{{r}^{2}}{{\sin }^{2}}\theta ={{r}^{2}}\left( {{\cos }^{2}}\theta +{{\sin }^{2}}\theta \right)={{r}^{2}}\times 1={{r}^{2}}. Also, we will take care of all types of calculation mistakes while solving the question.