Converting Polar Coordinates to Cartesian

The polar coordinates are defined in terms of $r$ and $\theta$, where $r$ is the distance of the point from the origin and $\theta$ is the angle made with the positive $x$-axis.

Clearly, using trigonometry, if the Cartesian coordinates are $(x,y),$ then

$$\begin{array}{c}&x = r \cos \theta, &y = r \sin \theta. \end{array}$$

If $3$ is the distance of a point from the origin and $\frac{\pi}{3}$ is the angle made with the positive $x$-axis, then what will be the Cartesian coordinates of the point?

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The Cartesian coordinates can be obtained from the distance $3$ of the point from the origin and the angle $\frac{\pi}{3}$ made with the positive $x$-axis. Using trigonometry, we have

$$\begin{aligned} x &= 3\cos\frac{\pi}{3} \\ &= \frac{3}{2} \\ y &= 3\sin\frac{\pi}{3} \\ &= \frac{3\sqrt{3}}{2}. \end{aligned}$$

Thus, the Cartesian coordinates of the point are $\displaystyle \left ( \frac{3}{2}, \frac{3\sqrt{3}}{2}\right ).$ $_\square$

Write the Cartesian coordinates $(2, 2\sqrt{3})$ as polar coordinates.

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Let $a$ denote the distance between the origin and the point $(2, 2\sqrt{3} ),$ then

$$a = \sqrt{ 2^2 + (2\sqrt{3})^2} = 4. \qquad (1)$$

Now, let $\theta$ denote the angle made with the positive $x$-axis, then we have

$$\begin{aligned} \sin \theta &= \frac{2\sqrt{3}}{4} = \frac{\sqrt{3}}{2} \\ \cos \theta &= \frac{2}{4} = \frac{1}{2}, \end{aligned}$$

which both imply $\theta=\frac{\pi}{3} .\qquad (2)$

Thus from $(1)$ and $(2),$ the polar coordinates of the Cartesian coordinates $(2, 2\sqrt{3} )$ are $$(a, \theta) = \left(4, \frac{\pi}{3}\right). \ _\square$$

What are the Cartesian coordinates of the point whose distance from origin is $4$ and the angle made with the positive $x$-axis is $\frac{\pi}{4} ?$

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Since the distance of the point from the origin is $4$ and the angle made with the positive $x$-axis is $\frac{\pi}{4} ,$ the Cartesian coordinates are

$$\begin{aligned} x &= 4\cos\frac{\pi}{4} = 2\sqrt{2} \\ y &= 4\sin\frac{\pi}{4} = 2\sqrt{2}. \end{aligned}$$

Thus, the answer is $\left(2\sqrt{2}, 2\sqrt{2} \right).$ $_\square$

Write the Cartesian coordinates $\left(\sqrt{3}, 1\right)$ as polar coordinates.

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Let $a$ denote the distance between the origin and the point $\left(\sqrt{3}, 1 \right),$ then

$$a = \sqrt{ (\sqrt{3})^2 + 1^2} = 2. \qquad (1)$$

Now, let $\theta$ denote the angle made with the positive $x$-axis, then we have

$$\begin{aligned} \sin \theta &= \frac{1}{2} \\ \cos \theta &= \frac{\sqrt{3}}{2}, \end{aligned}$$

which both imply $\theta=\frac{\pi}{6} . \qquad (2)$

Thus, from $(1)$ and $(2),$ the polar coordinates of the point $\left(\sqrt{3}, 1\right)$ in Cartesian plane are $$(a, \theta) = \left(2, \frac{\pi}{6} \right). \ _\square$$

What are the Cartesian coordinates of the point whose distance from origin is $10$ and the angle made with the positive $x$-axis is $\frac{\pi}{2} ?$

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Since the distance of point from the origin is $10$ and the angle made with the positive $x$-axis is $\frac{\pi}{2} ,$ the Cartesian coordinates are

$$\begin{aligned} x &= 10\cos\frac{\pi}{2} = 0 , \\ y &= 10\sin\frac{\pi}{2} = 10. \end{aligned}$$

Thus, the answer is $(0, 10).$ $_\square$

ParallelAngleBisector

The radius of the above circle is $10$ and the Cartesian coordinates of $A$ is $(5, 5\sqrt{3} ).$ If $\angle AOB$ is $60^\circ ,$ what are the Cartesian coordinates of $B?$

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From the cartesian coordinates of $A$ $(5,5\sqrt{3}) ,$ we can get the angle between $\overline {OA}$ and the positive $x$-axis using the polar coordinates of $A.$

Let $r\cos \theta$ and $r\sin \theta$ be the polar coordinates of $A,$ then since $r$ is $10,$ the angle $\theta$ is established as follows:

$$\begin{aligned} 10 \cos \theta &= 5 \\ \cos \theta &= \frac{1}{2}, \\ 10 \sin \theta &= 5\sqrt{3} \\ \sin \theta &= \frac{\sqrt{3}}{2}, \\ \Rightarrow \theta &= 60^\circ. \end{aligned}$$

Since the angle between $\overline {OA}$ and the positive $x$-axis is $60^\circ ,$ the angle between $\overline {OB}$ and the positive $x$-axis is $60^\circ + 60^\circ = 120^\circ.$

Hence, the Cartesian coordinates of $B$ are
$$\begin{aligned} x&= 10 \cos 120^\circ = 10\left ( -\frac{1}{2} \right ) = -5, \\ y&= 10 \sin 120^\circ = 10\left( \frac{\sqrt{3}}{2} \right ) = 5\sqrt{3}. \ _\square \end{aligned}$$