The idea of equations of any sort is an essential component of mathematics. Equations are the formulae which indicate the equality of two expressions. To write an equation, you write two expressions and connect them with the sign of equality (=). The method of expressing the relationship between two variables comprises framing an appropriate equation. Equations also help us in determining missing values of expressions following certain steps. From basic mathematics to advanced mathematics and all those places where there is an application of mathematics, equations play an extremely important role in expressing notions in a mathematical form.
Solving an equation refers to the method of finding the value(s) of the unknown variable(s) which makes the equality true. The variable whose value we try to find in the process is called “unknown” and its value(s) which satisfies the equality is called solution(s). An equation has two sides: the left-hand side (LHS) and the right-hand side (RHS). The LHS refers to the expression to the left of the equality sign, and the RHS refers to the expression to the right side of the equality sign.
An equation is analogous to a balanced weighing scale. Both sides must have equal quantities and if we are changing the quantities, then we should add or remove the same quantity on both sides to keep the equality intact. A balanced see-saw is another example which proves the point.
There are many types of equations and their operations, and there are ways of converting one into the other’s form and vice-versa. Rectangular equations and polar equations are two of the many types of equations. Here we will see how we can convert a rectangular equation to a polar equation using a rectangular to polar equation calculator.
Rectangular to polar equation calculator
Rectangular equations are one of the most important and common types of equations. A rectangular equation is an equation which we can graph on a Cartesian plane. It contains the variables x and y, which represent the two axes, in any functional form such that we can represent the equation in a graphical form. For example, y = 44 + 6x is a rectangular equation. For those who do not know, in mathematics, a Cartesian plane is a two-dimensional coordinate plane made by the intersection of the x and y-axis. The point where the two axes intersect is the origin.
Polar equations (or polar curves) describe the relationship between the distance from the pole (origin) to a point on a curve and the counterclockwise angle a point makes on a curve. We may represent the distance from the pole (origin) to a point on a curve by r and the counterclockwise angle by θ (theta). The polar coordinate system is a two-dimensional coordinate system in which we determine every point by a distance from a reference point and an angle from a reference direction.
To convert rectangular equations to polar equations, we can use a rectangular to polar equation calculator. A rectangular to polar equation calculator gives us a suitable process following which we can transform any transformable rectangular equation into a polar equation. In essence, following the rectangular to polar equation calculator, we rewrite an equation in terms of r and θ (theta). We replace x and y with suitable expressions and rewrite the equation.
Rectangular to polar equation calculator y=x^2
Imagine a point P on a Cartesian plane with the coordinates (x,y). X is the perpendicular distance of the point from the vertical axis, and y is the perpendicular distance of the point from the horizontal axis. If we draw a right-angled triangle with x as the base and y as the height, we get the distance between P and origin as the hypotenuse length. In such a case, if the point is P, the point at a distance of x from the origin is A, and the origin is O, then there is a triangle POA. So, our base is OA (x), height is AP (y), and the hypotenuse is OP (r). Let the triangle POA have the angle POA as θ. Following such specifications and our knowledge of basic trigonometry, we know PA/PO = sin θ. From here, we can rewrite it as
AP/OP = sin θ
=> AP = x = r(sin θ).
Thus, in our equation, we may rewrite x as r(sin θ).
Similarly, for y, we know
OA/OP = cos θ
=> OA = y = r(cos θ).
Therefore, we can rewrite y as r (cos θ).
So, following this method of rectangular to polar equation calculator, we have learnt how to rewrite x and y, the variables of a rectangular equation, in terms of r and θ, that is, polar form. Now let us put this into application.
Suppose our rectangular equation looks like y = x2. How do we rewrite this in the polar form? We will use the conversion technique we just learnt in the above rectangular to polar equation calculator.
y = x2
=> r (cos θ) = r2(sin θ)2
=> r = (cosec θ)(cot θ).
Therefore, y =x2 in its polar form is r = (cosec θ) (cot θ).
Rectangular to polar equation calculator wolfram
The internet has the answer to almost any question one has. But how does it answer our questions? The simplest answer is that people put up answers and algorithms over the internet which finds our answers. These are uploaded on internet sites and our search engine (Google, Bing, and Yahoo!) finds the site which serves our purpose best. Then you can write your question there in a particular format and their website will yield the results. This is very helpful for everyone, especially students and researchers. They can use the materials readily available over the web for their benefit immensely. Everything is just a couple of minutes away.
Wolfram Alpha is one such website. It is a computational knowledge engine and answer engine which uses algorithms to solve questions, including mathematical ones. Wolfram Alpha uses externally sourced data to compute answers for factual queries. You can find the solution for every common mathematical question and transformation, even the rectangular to polar equation calculator of Wolfram Alpha. If you wish to use the internet to find the polar form of a rectangular equation, Wolfram Alpha would be an excellent choice although there certainly are other sources too. Furthermore, since the process is very easy, you should do the conversions manually, at least for simple equations. The idea behind such an exercise is to develop an understanding of how the thing is done under various cases, that is, forms of the rectangular equation.
To launch the page of Wolfram Alpha which will help you with the calculation, you can simply search “rectangular to polar equation calculator wolfram”. “Rectangular to polar equation calculator wolfram alpha” also works. The search results lead you directly to that particular page of Wolfram Alpha where you can put your rectangular equation and generate its polar form.
Rectangular to polar equation calculator Mathway
Mathway is another such website which you can use for mathematics. It was developed in 2002 by two middle school friends as an online problem solver. As you can easily guess by now, Mathway works in a very similar way to Wolfram Alpha. You enter your mathematical expression and apply the mathematical tool to achieve the desired results. Over time, Mathway has become the world’s first fully automated online mathematics tutor too. Besides its website, Mathway is available as a mobile application too.
You can use Mathway as a rectangular to polar equation calculator too in place of Wolfram Alpha. On the website or application, go to the relevant page and enter your equation in its rectangular form, run the analysis, and get its polar form. So now you have a fair idea of how mathematics has been made extremely easy and accessible for everyone. However, it is strongly encouraged that students should learn mathematical applications and tools manually first. Once they have the theory’s idea, they can interpret the results very easily or even solve the problems in case the internet is not accessible. They will also be able to detect any inconsistencies in their mathematical model. In most cases, the website would not help you detect any problem with the question itself apart from showing that the question is not solvable.
Because there are many things that mathway can do, it can often be very tedious to manually locate the function or page where you can put up your question for an answer. Just type “rectangular to polar equation calculator mathway” and your search engine will lead you directly to that page of mathway which looks after this problem.
Rectangular to polar coordinates calculator
The Cartesian or rectangular coordinate system is the system wherein a point is given a unique “address” on a Cartesian plane in terms of its distance from the two axes which intersect each other perpendicularly.
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We denote the point’s location by two numbers within parentheses separated by a comma. The first number and the second number are the perpendicular distances of the point from the y-axis and the x-axis respectively. If the coordinates are, say, (5, 3), then is 5 units away from the y-axis toward the right of the y-axis, and 3 units away towards the top from the x-axis. Note that the coordinates can have negative numbers too. If the coordinates are (-5,-3), then the same point is the 5 units to the left of y-axis and 3 units below the x-axis. The point can be a combination of negative and positive numbers also. In essence, the point is present anywhere among the four “quadrants” on the Cartesian plane.
Another type of coordinate system in mathematics is the polar coordinate system. Here we represent a point’s location in terms of an angle and its distance from the origin. The point is in terms of two numbers in parentheses here also, the first being the distance from the origin (pole in the polar coordinate system) and the second the angle the line connecting the point and the pole makes with the horizontal axis. So, if the polar coordinates are (7,77°), the point is 7 units away from the pole and the line connecting the point and the pole makes an angle of 77° with the horizontal axis.
Moreover, just like a rectangular to polar equation calculator, we have a rectangular to polar coordinates calculator too. We can use it to convert rectangular coordinates to polar coordinates.
Cartesian to polar equation calculator x^2+y^2=9
We know how to convert rectangular equations to polar equations. Just replace x by r (cos θ) and y by r(sin θ). The same thing is applicable for the equation x2 + y2 = 9 also.
x2 + y2 = 9
=> r2 (cos θ) 2 + r2 (sin θ) 2 = 9
=> (cos2 θ + sin2 θ) r2 = 9
=> r2 = 9
=> r = 3.
Explanation: First we replace x and y with their counterparts in terms of r and θ (theta). Then we take r2 common and add cos2 θ and sin2 θ. Trigonometry tells us their sum is 1, so we are left with r2 on the LHS and 9 on the RHS. If we take the square root of both sides, we get r = 3. This is our equation.
Thus, this is how we transform x2 + y2 = 9 to its polar form using the rectangular to polar equation calculator.
Polar to rectangular equation calculator
Just the way you can have questions asking you to transform rectangular equations to polar equations; you can have questions asking you to convert polar equations to rectangular equations also. In such a case, you need a polar to rectangular equation calculator in place of a rectangular to polar equation calculator.
While converting rectangular equations to polar equations, we replaced x with r (cos θ) and y with r(sin θ). Polar equations will be in terms of trigonometric ratios and r, the distance of the point from the pole. What do we do here? We bring the trigonometric ratios in terms of sin θ and cos θ and have them multiplied with r. This transformation is important to bring the components of the polar equation to a form which we can convert to functions of x and y.
Suppose your question is to convert r = cosec θ to its rectangular form. Follow the steps below:
r = 3(cosec θ)
=> r = 3/ (sin θ)
=> (sin θ) r = 3
=> y = 3.
Explanation: We know cosec θ is the reciprocal of sin θ. So, we convert cosec θ to sin θ. Then we multiplied both sides by sin θ. This helps us get the standard r(sin θ) on the LHS. Lastly, we substitute y in place of r(sin θ) and get the rectangular form of the polar equation. Therefore, the rectangular form of r = 3(cosec θ) is y = 3.
Rectangular to polar equation calculator FAQs
How do you convert rectangular form to polar form?
To convert rectangular equation to polar equation, replace x with r (cos θ) and y with r(sin θ). This will give the polar form of your rectangular equation.
How do you write numbers in polar form?
To write any number in its polar form, we use a set of formulae. They are:
x = r(cos θ), y = r(sin θ), and r = √(x2 + y2). Then, z = r(cos θ + isin θ).
Here, r is the distance of the point from the pole and z is the length of the line segment connecting the point and the origin. θ is the angle the line segment connecting the point and the origin makes with the horizontal axis.
How do you convert rectangular to polar on MATLAB?
MATLAB is software for mathematical and statistical analyses. This means we can use MATLAB as a rectangular to polar equation calculator and rectangular to polar coordinates calculator. [Theta, rho] = cart2pol(x, y) will transform our Cartesian (rectangular coordinates) to polar coordinates. The Cartesian coordinates are (x, y) and the polar coordinates are [ theta , rho ].
How does cart2pol work MATLAB?
[x , y ] = cart2pol (theta , rho ) transforms the corresponding Cartesian coordinates to polar coordinates.