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The polar coordinate system is defined as a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle is taken from a reference direction. The pole is a reference point. The polar axis is the line segment ray that extends from the pole in the reference direction. The origin is referred to as a pole in the polar coordinate system. Here we will learn about cartesian conversion to polar coordinates and solve some important questions.
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Key Takeaways: Coordinate System, Cartesian, Polar, formula, conversion
Also check: Quadrilateral Formula
Polar Coordinate Formula
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We can use the formula to generate an infinite number of polar coordinates for a single coordinate point. The formula is written as follows:
(r, θ + 2πn) or (-r, θ + (2n+1)π)
In this case, n is an integer.
If measured counterclockwise, the value of will be positive, whereas it will be negative if measured clockwise. Similarly, if you lay off the terminal side of, the value of r will be positive, whereas if you lay off the prolongation through the origin from the terminal side of, the value of r will be negative. The side where the angle begins is known as the initial side, and the ray where the angle measurement ends is known as the terminal side.
The video below explains this:
Complex Numbers Detailed Video Explanation:
Also check: Difference between Sequence and Series
Point Plotting in a Polar Coordinate System
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3,60 and 4,210 are the two points.
There are two polar coordinates in a two-dimensional Polar Coordinate system: r and, which represent the radial distance from the pole and the angular coordinate, which represents the anticlockwise angle from the 0° ray, respectively. On the Cartesian coordinate plane, it is also known as the positive x-axis.
For a better understanding, consider some polar coordinates examples.
Consider that the polar coordinates (3,60°) are plotted on the 60° ray as a point 3 units from the pole. Because the negative radial distance is measured as a positive distance on the opposite ray (240° 180° = 60°), the coordinates (3,240°) will also be plotted exactly at this point.
Another important feature of the Polar Coordinate System that the Cartesian coordinate system lacks is the ability to express a single point with an infinite number of different coordinates. In most cases, the point (r,) can also be represented as (r, n 360°) or (r, (2n + 1)180°).
where n is a positive integer If a point's r coordinate is 0, it will be located at the pole regardless of the coordinate.
Also check:
Converting from Cartesian to Polar Coordinate Systems
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To convert a point in Cartesian Coordinates (x,y) to Polar Coordinates (r,), we must first solve a right triangle with two known sides.
For example consider the points (12,5)
We can use Pythagoras theorem to find the hypotenuse
r2 = 122 + 52
r = √(122 + 52)
r = √(144+25)
r = √(169)
r = 13.
Now, to find the angle, we will use the tangent function.
Tan ( θ ) = 5/12
θ = tan-1(5/12) = 22.6°
Therefore, point (12, 5) in the cartesian coordinate system will be (13, 22.6°) in the Polar Coordinate System.
Also check: Differentiation and Integration Formula
Polar Coordinate System to Cartesian Coordinate System
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Conversion is relatively simple to convert the polar coordinate system to Cartesian coordinate systems. We simply take the cosine of to find the corresponding Cartesian x coordinate and the sine of to find y.
It is simple to determine polar coordinates from a given pair of Cartesian coordinates using basic trigonometry.
r = \(\sqrt{x^2 + y^2}\)
θ = tan-1 (y/x)
Things to Remember
- In polar coordinates, rather than using signed distances along the two coordinate axes, specify the location of a point P in the plane by its distance r from the origin and the angle formed by the line segment from the origin to P and the positive x-axis.
- Using the formula, we can generate an infinite number of polar coordinates for a single coordinate point. (r, +2n) or (-r, +(2n+1)), where n is a positive integer.
- When measured counterclockwise, the value of is positive.
- When measured clockwise, the value of is negative.
- If r is laid off at the terminal side of, it has a positive value.
- If laid off at the prolongation through the origin from the terminal side of, the value of r is negative.
Also check: Calculus Formula
Sample Questions
Ques. What exactly is a Polar Curve? (2 marks)
Ans. A polar curve is a shape that is created primarily using the polar coordinate system. Polar curves are defined by points at varying distances from the origin, i.e., the pole, depending on the angle measured off the positive x-axis. Polar curves can be used to describe both familiar Cartesian shapes like ellipses and some unfamiliar shapes like cardioids and lemniscates.
Ques. What are the Polar Coordinates System's Real-World Applications? (2 marks)
Ans. The polar coordinates system (r and ) can be extremely useful for calculating the equations of motion from a variety of mechanical systems for physicists. The Lagrangian and Hamiltonian techniques can be used to determine the dynamics of objects moving in a circle.
However, the advantage of using a polar coordinates system is that it will greatly simplify things by producing neat and understandable derived equations. Aside from the mechanical system, polar coordinates can also be used in 3D, which aids in the calculation of many fields such as electric fields, magnetic fields, and temperature fields. So, in general, the polar coordinate system simplifies calculations for physicists and engineers.
Ques. In polar form, represent the complex number z =1+ √3. (3 marks)
Ans. let z =1+ i √3 = r(cos θ + i sin θ)
r=| z | = (a2 + b2 )1/2 = ((1)2 + (√3)2)1/2 = 2
Comparing real parts of z =1+ i √3 = r(cos θ + i sin θ) = 2(cos θ + i sin θ)
1 = 2 cosθ
Or, cosθ = ½
Or, cosθ = π/3
Therefore, polar representation will be z = r(cosθ + i sinθ) = 2(cos π/3 + i sin π/3)
Ques: Express the given complex number in the form a + ib: (5i) ( – \(\frac{3}{5}\)i) (3 marks)
Ans:
Ques: Find the coordinates of the point on y-axis which are at a distance of 52–√52 from the point P(3,-2,5) (3 marks)
Ans: For the point to be on the x-axis the y-coordinate and z-coordinate become zero.
Let the point on y-axis at a distance of 5√2 from point P(3,-2,5)P(3,-2,5) be A(0,b,0)A(0,b,0),
We have, AP = 5√2
Using distance formula,
AP2 = 50
(3 – 0)2 + (- 2 – b)2 + (5 – 0)2 = 50
9 + 4 + b2+ 4b + 25 = 50
b2 + 4b - 12 = 0
b2 + 6b - 2b – 12 = 0
(b + 6) (b – 2) = 0
b = – 6b or b = 2
The coordinate of the points is (0,2,0)(0,2,0) and (0,-6,0)
Ques. Three vertices of a parallelogram ABCD are A (3,-1,2), B (1,2,-4) and C (-1,1,2). Find the coordinates of the fourth vertex. (3 marks)
Ans. We are given the three vertices of a parallelogram ABCD are A(3,-1,2), B(1,2,-4) and C(-1,1,2).
Let the coordinates of the fourth vertex of the parallelogram ABCD be D(x,y,z).
According to the property of parallelogram, the diagonals of the parallelogram bisect each other.
In this parallelogram ABCD, AC and BD at point O.
So,
Midpoint of AC = Midpoint of BD
(3-12, -1+12, 2+22) = (x+12, y+12, z-42)
(1,0,2) = (x+12, y+12, z-42)
x+12 = 1
y+22 = 0
z-42 = 2
We get, x = 1, y = 2 and z = 8
Therefore, the coordinates of the fourth vertex of the parallelogram ABCD are D(1,-2,8).
Ques: A point R with x-coordinate 4 lies on the line segment joining the points P(2,-3,4) and Q(8,0,10). Find the coordinates of the point R. (3 marks)
Ans: Let R is point which divides the line segment PQin the ratio k:1.
Using section formula,
(k(8)+ 2k + 1, k(0) - 3k + 1, k(10) + 4k + 1) = (8k + 2k + 1, −3k + 1, 10k + 4k + 1)
The value of x-coordinate of the point R is 4,
8k + 2k + 1 = 4
8k + 2 = 4k + 4
4k = 2
k = 12
So, the coordinates of the point R are,
(4,-312+1,10(12)+412+1) = (4,-2,6)
Ques: Find the polar coordinates of (-3, 0) (2 marks)
Ans: We are given the Cartesian coordinates(−3, 0). So we have x = −3<0, y = 0. Then we use conversion formula to have
r=√ (−3)2+(0)2 = 9
θ= tan-1 0/−3 + π
tan−1 0 + π = π
So the required polar coordinate is (r,θ)=(1,π)
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