Multiple Representations Of Polar Coordinates Calculator (2024)

FAQs

Are there multiple ways to represent polar coordinates? ›

In the polar coordinate system, points may be expressed in multiple ways. Point A can be expressed as (1, π/6) or also as (-1, 7π/6), or also as (1, 13π/6) or (-1, 19π/6), etc… Therefore, a point with polar coordinates (r, θ) can also be represented by: (r, θ+2kπ) or (-r, θ+π+2kπ) for any integer k.

In the polar coordinate system, the origin is called a pole. Here, instead of representing the point as (x, y), we can express it as a polar coordinate (r, θ). Where the value of r can be negative. The value of angle changes based on the quadrant in which the r lies.

To convert from Cartesian coordinates to polar coordinates: r = x 2 + y 2 . Since tan θ = y x , θ = tan − 1 ( y x ) . So, the Cartesian ordered pair converts to the Polar ordered pair ( r , θ ) = ( x 2 + y 2 , tan − 1 ( y x ) ) .

To multiply complex numbers in polar form, multiply the magnitudes and add the angles. To divide, divide the magnitudes and subtract one angle from the other.

The rectangular coordinates of a point are unique, but the polar coordinates are not unique. Every point has infinitely many polar coordinate representations. and there are infinitely many other ways to represent this point. Problem: Graph the point that has polar coordinates (4,).

The polar form of a complex number z = x + iy with coordinates (x, y) is given as z = r cosθ + i r sinθ = r (cosθ + i sinθ). The abbreviated polar form of a complex number is z = rcis θ, where r = √(x² + y²) and θ = tan^-1 (y/x).

Our conversion formula is x=rcosθy=rsinθ. To go the other direction, one can use the same right triangle. Since r is the distance from the origin to (x,y), it is the magnitude r=√x2+y2. Alternatively, from the equation (1), one can calculate directly that x2+y2=r2cos2θ+r2sin2θ=r2(cos2θ+sin2θ)=r2.

To obtain the polar form, we will use the relationships between ( x , y ) ( x , y ) and ( r , θ ) . ( r , θ ) . Since x = r cos θ x = r cos θ and y = r sin θ , y = r sin θ , we can substitute and solve for r .

and thus the polar co-ordinates are (√2, −π/4). To convert from polar co-ordinates to Cartesian co-ordinates, use the equations x = r cos θ , y = r sin θ . To convert from Cartesian co-ordinates to polar co-ordinates, use the equations r2 = x2 + y2 , tan θ = y x .

What is the expression for polar coordinates? ›

Uniqueness of polar coordinates

Therefore, the same point (r, φ) can be expressed with an infinite number of different polar coordinates (r, φ + n × 360°) and (−r, φ + 180° + n × 360°) = (−r, φ + (2n + 1) × 180°), where n is an arbitrary integer. Moreover, the pole itself can be expressed as (0, φ) for any angle φ.

Among the best known of these curves are the polar rose, Archimedean spiral, lemniscate, limaçon, and cardioid. For the circle, line, and polar rose below, it is understood that there are no restrictions on the domain and range of the curve.

Thus, the polar coordinates (r, θ) and (r, θ + 2Kπ) for any integer K represent the same complex number. Thus, the polar representation is not unique; by convention, a unique polar representation can be obtained by requiring that the angle given by a value of θ satisfying 0 ≤ θ < 2π or -π < θ ≤ π.

There are five classic polar curves: cardioids, limaҫons, lemniscates, rose curves, and Archimedes' spirals.