4.2: Polar and Cartesian Transformation
You will see during this lesson that points can be converted from rectangular form to polar form with a little Algebra and Trigonometry.
Can the equation of a shape be converted also? How about a circle, for instance?
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 Khan Academy: Polar Coordinates 2
Guidance
Polar Form to Rectangular Form
Sometimes a problem will be given as a coordinate in polar form but rectangular form may be needed.
To transform the polar point \begin{align*}\left (4, \frac{3\pi}{4} \right )\end{align*} into rectangular coordinates: first identify (r, θ)
r = 4 and \begin{align*}\theta = \frac{3\pi}{4}\end{align*}.
Second, draw a vertical line from the point to the polar axis (the horizontal axis). The distance from the pole to where the line you just drew intersects the polar axis is the x value, and the length of the line segment from the point to the polar axis is the y value.
These distances can be calculated using Trigonometry:
x = r cos θ and y = r sin θ
\begin{align*}x = 4 \ \mbox{cos} \ \frac{3\pi}{4}\end{align*} and \begin{align*}y = 4 \ \mbox{sin} \ \frac{3\pi}{4}\end{align*} or \begin{align*}x = 2 \sqrt{2} \ \ y = 2\sqrt{2}\end{align*}
\begin{align*}\left (4, \frac{3\pi}{4} \right )\end{align*} in polar coordinates is equivalent to \begin{align*}(2\sqrt{2}, \ 2\sqrt{2})\end{align*} in rectangular coordinates.
Rectangular Form to Polar Form
Going from rectangular coordinates to polar coordinates is also possible, but it takes a bit more work. Suppose we want to find the polar coordinates of the rectangular point (2, 2). To begin doing this operation, the distance that the point (2, 2) is from the origin (the radius, r) can be found by
\begin{align*}r = \sqrt{x^2 + y^2}\end{align*}
\begin{align*}r = \sqrt{2^2 + 2^2}\end{align*}
\begin{align*}r = \sqrt{8} = 2\sqrt{2}\end{align*}
The angle that the line segment between the point and the origin can be found by
\begin{align*}\mbox{tan} \ \theta = \frac{y}{x}\end{align*}
\begin{align*}\mbox{tan} \ \theta = \frac{2}{2}\end{align*}
\begin{align*}\mbox{tan} \ \theta = 1\end{align*}
\begin{align*}\theta = \mbox{tan}^{1} 1\end{align*}
\begin{align*}\theta = \frac{\pi}{4}\end{align*}
Since this point is in the first quadrant (both the x and y coordinate are positive) the angle must be 45^{o} or \begin{align*}\frac{\pi}{4}\end{align*} radians. It is also possible that when tan θ = 1 the angle can be in the third quadrant, or \begin{align*}\frac{5\pi}{4}\end{align*} radians. But this angle will not satisfy the conditions of the problem, since a third quadrant angle must have both x and y negative.
Note: when you use using \begin{align*}\mbox{tan} \ \theta = \frac{y}{x}\end{align*} to find the measure of θ you should consider, at first, the quotient \begin{align*}\mbox{tan} \ \theta = \left  \frac{y}{x} \right \end{align*} and find the first quadrant angle that satisfies this condition. This angle will be called the reference angle, denoted θ_{ref}. Find the actual angle by analyzing which quadrant the angle must be given the signs of x and y.
Example A
Transform the polar coordinates \begin{align*}\left (2, \frac{11\pi}{6} \right )\end{align*} to rectangular form
Solution
\begin{align*}r = 2\end{align*} and \begin{align*}\theta = \frac{11\pi}{6}\end{align*}
\begin{align*}x = r \ \mbox{cos} \ \theta\end{align*} and \begin{align*}y = r \ \mbox{sin} \ \theta\end{align*}
\begin{align*}x = 2 \ \mbox{cos} \ \frac{11\pi}{6}\end{align*} and \begin{align*}y = 2 \ \mbox{sin} \ \frac{11\pi}{6}\end{align*} or \begin{align*}x = 3\sqrt{2} \ \ y = 1\end{align*}
\begin{align*}\left (2, \frac{11\pi}{6} \right )\end{align*} is equivalent to \begin{align*}(3\sqrt{2}, 1)\end{align*} or in decimal form, approximately \begin{align*}(4.342, 1)\end{align*}
Example B
Find the Polar coordinates for \begin{align*}(3, 3\sqrt{3})\end{align*}
\begin{align*}x = 3\end{align*} and \begin{align*}y = 3\sqrt{3}\end{align*}
Solution
Draw a right triangle in standard form. Find the distance the point is from the origin and the angle the line segment that represents this distance makes with the +x axis:
\begin{align*}r = \sqrt{3^2 + (3\sqrt{3})^2}\end{align*}
\begin{align*}= \sqrt{9 + 27}\end{align*}
\begin{align*}= \sqrt{36}\end{align*}
\begin{align*}= 6\end{align*}
And for the angle,
\begin{align*}\mbox{tan} \ \theta_{ref} = \left  \frac{(3\sqrt{3})}{3} \right \end{align*}
\begin{align*}\mbox{tan} \ \theta_{ref} = \sqrt{3}\end{align*}
\begin{align*}\theta_{ref} = \mbox{tan}^{1} \sqrt{3}\end{align*}
\begin{align*}\theta_{ref} = \frac{\pi}{3}\end{align*}
So, \begin{align*}\theta_{ref} = \frac{\pi}{3}\end{align*} and we can look at the signs of x and y  (+, )  to see that \begin{align*}\theta = \frac{5\pi}{3}\end{align*} since it is a 4^{th} quadrant angle.
The rectangular point \begin{align*}(3, 3\sqrt{3})\end{align*} is equivalent to the polar point \begin{align*}\left (6, \frac{5\pi}{3} \right )\end{align*}.
Recall that when solving for θ, we used
\begin{align*}\mbox{tan} \ \theta = \left  \frac{(3\sqrt{3})}{3} \right \end{align*} or \begin{align*}\mbox{tan} \ \theta = \sqrt{3}\end{align*}
We found
\begin{align*}\theta = \frac{5\pi}{3}\end{align*}. BUT, θ could also be \begin{align*}\theta = \frac{2\pi}{3}\end{align*}. You must examine the signs of each coordinate to see that the angle must be in the fourth quadrant in rectangular units or between \begin{align*}\frac{3\pi}{2}\end{align*} and 2π in polar units. Of the two possible angles for θ, only \begin{align*}\frac{5\pi}{3}\end{align*} is valid. Note that when you use tan^{1} on a calculator you will always get an answer in the range \begin{align*} \frac{\pi}{2} \le \theta \le \frac{\pi}{2}\end{align*}.
Example C
Convert the following rectangular coordinates to polar coordinates
a. \begin{align*}(3, 3\sqrt{3})\end{align*}
b. \begin{align*}(2, 2)\end{align*}
Convert the following polar coordinates to rectangular coordinates:
c. \begin{align*}\left (4, \frac{2\pi}{3} \right )\end{align*}
d. \begin{align*}\left (1, \frac{5\pi}{6} \right )\end{align*}
Solutions
a. \begin{align*}(6,66^\circ)\end{align*}
b. \begin{align*}(2\sqrt{2}, 225^\circ)\end{align*}
c. \begin{align*}(2, 2\sqrt{3})\end{align*}
d. \begin{align*}\left ( \frac{\sqrt{3}}{2},  \frac{1}{2} \right )\end{align*}
Concept question wrap up Equation of a circle x^{2} + y^{2} = k^{2} is the equation of a circle with a radius of k in rectangular coordinates. The equation of a circle is extremely simple in polar form. In fact, a circle on a polar graph is analogous to a horizontal line on a rectangular graph! You can transform this equation to polar form by substituting the polar values for x, y. Recall x = r cos θ and y = r sin θ (r cos θ)^{2} + (r sin θ)^{2} = k^{2}, square the terms: r^{2} cos^{2} θ + r^{2} sin^{2} θ = k^{2}, factor the r^{2} from both terms on the left: r^{2} (cos^{2} θ + sin^{2} θ) = k^{2} recall the identity: cos^{2} θ + sin^{2} θ = 1 r^{2} = k^{2} Therefore: \begin{align*}r = \pm k\end{align*} is an equation for a circle in polar units. When r is equal to a constant, the polar graph is a circle.


Vocabulary
Quadrants are each of the four "corners" of a graph.
The sine of an angle may be found by dividing the length of the side opposite the angle in question by the length of the hypotenuse.
The cosine of an angle may be found by dividing the length of the side adjacent (next to) the angle in question by the length of the hypotenuse.
The tangent of an angle may be found by dividing the length of the side opposite the angle in question by the length of the side adjacent to it.
Guided Practice
1) Change \begin{align*}(3, 45^o)\end{align*} to rectangular coordinates.
2) Change \begin{align*}\left(3, \frac{\pi}{4}\right)\end{align*} to rectangular coordinates.
3) Express the equation in rectangular form: \begin{align*}r = 6 cos \theta\end{align*}
4) Express the equation in rectangular form: \begin{align*}r = 6\end{align*}
Answers
1) To change the description of the point to rectangular, first find the xvalue, then the yvalue, as follows:
xcoordinate:
 \begin{align*}x = r \cdot cos\theta\end{align*} : the \begin{align*}x\end{align*} coordinate is found by multiplying \begin{align*}r\end{align*} by the cosine of \begin{align*}\theta\end{align*}
 \begin{align*}x = 3 \cdot cos (45^o)\end{align*} : substitute the given information for \begin{align*}r\end{align*} and \begin{align*}\theta\end{align*}
 \begin{align*}x = 2.121\end{align*}
ycoordinate:
 \begin{align*}y = r \cdot sin\theta\end{align*} : the \begin{align*}y\end{align*} coordinate is found by multiplying \begin{align*}r\end{align*} by the sine of \begin{align*}\theta\end{align*}
 \begin{align*}y = 3 \cdot sin (45^o)\end{align*} : substitute the given information for \begin{align*}r\end{align*} and \begin{align*}\theta\end{align*}
 \begin{align*}y = 2.121\end{align*}
\begin{align*}\therefore (2.121, 2.121)\end{align*} is the location of \begin{align*}(3, 45^o)\end{align*} in rectangular form.
2) To change \begin{align*}\left( 3, \frac{\pi}{4} \right)\end{align*} to rectangular coordinates, use the same process as Q #1:
xcoordinate:
 \begin{align*}3 cos \left( \frac{\pi}{4} \right)\end{align*} : substituting the values from the problem into \begin{align*}x = r \cdot cos\theta\end{align*}
 \begin{align*}3 cos 45\end{align*} : \begin{align*}\frac{\pi}{4} = 45^o\end{align*}
 \begin{align*}x = 2.121\end{align*}
ycoordinate
 \begin{align*}3 sin \left( \frac{\pi}{4} \right)\end{align*} : substituting the values from the problem into \begin{align*}y = r \cdot sin\theta\end{align*}
 \begin{align*}3 sin 45\end{align*} : \begin{align*}\frac{\pi}{4} = 45^o\end{align*}
 \begin{align*}y = 2.121\end{align*}
\begin{align*}\therefore (2.121, 2.121)\end{align*} is the location of \begin{align*}\left( 3, \frac{\pi}{4} \right)\end{align*} in rectangular form.
3) To express \begin{align*}r = 6 cos \theta\end{align*} in rectangular form:
 \begin{align*}r^2 = 6 r cos \theta\end{align*} : multiply both sides by \begin{align*}r\end{align*}
 \begin{align*}x^2 + y^2 = 6x\end{align*} : Using \begin{align*}x^2 + y^2 = r^2\end{align*} and \begin{align*}x = r cos \theta\end{align*}
\begin{align*}\therefore x^2 + y^2 = 6x\end{align*} is the equation in rectangular form.
4) This one is easy:
 \begin{align*}r = 6\end{align*} is the polar form of the equation for a circle
 \begin{align*}r^2 = 6^2\end{align*} : square both sides
 \begin{align*}x^2 + y^2 = 36\end{align*} : Using \begin{align*}x^2 + y^2 = r^2\end{align*} and simplifying
\begin{align*}\therefore x^2 + y^2 = 36\end{align*} is the equation in rectangular form.
Practice
 How is the point with polar coordinates \begin{align*}(5, \pi)\end{align*} represented in rectangular coordinates?
Plot each point below in polar coordinates (r, θ). Then write the rectangular coordinates (x, y) for the point.
 \begin{align*}(3, 60^o)\end{align*}
 \begin{align*}\left( 10, \frac{\pi}{3} \right)\end{align*}
 \begin{align*}\left( 15, {\pi} \right)\end{align*}
The rectangular coordinates (x, y) are given. For each question: a) Find two pairs of polar coordinates (r, θ), one with r > 0 and the other with r < 0. b) Express θ in radians, and round to the nearest hundredth.
 \begin{align*}(5, 5)\end{align*}
 \begin{align*}(0, 10)\end{align*}
 \begin{align*}(8, 6)\end{align*}
Transform each polar equation to an equation using rectangular coordinates. Identify the graph, and give a rough sketch or description of the sketch.
 \begin{align*}\theta =\frac{\pi}{10}\end{align*}
 \begin{align*}r = 8\end{align*}
 \begin{align*}r sin \theta = 7\end{align*}
 \begin{align*}r cos \theta = 3\end{align*}
Transform each rectangular equation to an equation using polar coordinates. Identify the graph, and give a rough sketch or description of the sketch.
 \begin{align*}x^2 + y^2  2x = 0\end{align*}
 \begin{align*}y = \sqrt3 x\end{align*}
 \begin{align*}y = 5\end{align*}
 \begin{align*}xy = 15\end{align*}
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Term  Definition 

Tangent  The tangent of an angle in a right triangle is a value found by dividing the length of the side opposite the given angle by the length of the side adjacent to the given angle. 
cosine  The cosine of an angle in a right triangle is a value found by dividing the length of the side adjacent the given angle by the length of the hypotenuse. 
polar coordinates  Polar coordinates describe locations on a grid using the polar coordinate system. The location of each point is determined by its distance from the pole and its angle with respect to the polar axis. 
polar form  The polar form of a point or a curve is given in terms of and and is graphed on the polar plane. 
quadrant  A quadrant is onefourth of the coordinate plane. The four quadrants are numbered using Roman Numerals I, II, III, and IV, starting in the topright, and increasing counterclockwise. 
Quadrants  A quadrant is onefourth of the coordinate plane. The four quadrants are numbered using Roman Numerals I, II, III, and IV, starting in the topright, and increasing counterclockwise. 
rectangular coordinates  A point is written using rectangular coordinates if it is written in terms of and and can be graphed on the Cartesian plane. 
rectangular form  The rectangular form of a point or a curve is given in terms of and and is graphed on the Cartesian plane. 
sine  The sine of an angle in a right triangle is a value found by dividing the length of the side opposite the given angle by the length of the hypotenuse. 
Image Attributions
Here you will learn how to convert points and expressions between polar form and rectangular (Cartesian coordinate) form.