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Convert r sin θ = 4 to Rectangular Form

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A conic section is the branch of Mathematics, which deals with the study of curves formed by the intersection of a cone with a plane. The intersection between the cone and the plane can vary differently based on the point and angle of intersection. When considering the equations in Conic Section, the two most important ones are Polar equations and Rectangular Equations. In this article, we will discuss both polar and rectangle equations along with their conversion. 

Polar Equation 

Equations involving the relation between angle θ and distance ‘r’ fall under this category. Here, ‘r’ denotes the distance between the Origin (also called Pole) and a point on the curve and θ denotes the clockwise angle made by the origin, the positive side of the x-axis, and a point on a given mathematical curve.

x = r cos θ

y = r sin θ

Rectangular Equation 

Equations involving variables fall under this category of Rectangular Equations. There is no involvement of angle in such equations, and they are simply constants and variables with suitable mathematical operators. Rectangular equations can be easily represented as graphs on Cartesian Plane.

r2 = x2 + y2

Converting Polar equation to rectangle equation

Conversion of Polar to Rectangular form is easy. To convert polar to rectangular form, follow the below steps:

Step 1: List down the polar equations with r, θ angle, and variables x, y 

x = r cos θ (eq1)

y = r sin θ (eq2)

Step 2: Careful consider the situation, where theta angle can be eliminated.

Squaring eq1 we get:

x2 = r2 cos2θ (eq3)

Similarly, squaring eq 2 we get:

y2 = r2 sin2θ (eq4)

Step 3: Perform Mathematical addition as needed depending on the trigonometric equation involving ‘Sin theta’ and ‘Cos theta’

Adding eq3 and eq4

x2 + y2 = r2 cos2θ + r2 cos2θ

we already know, sin2θ + cos2θ = 1

r2 = x2 + y2

We see that theta gets eliminated from combining the equations, and thus in the above manner, we are able to convert the Polar equation to a rectangular equation.

Convert Polar equation r sinθ = 4 to rectangular form

Solution:

Given r sin θ = 4 …(1)

As we know that 

y = r sin θ

So, in the equation (1) replace r sinθ by y

y = 4

We conclude that the rectangular form of r sinθ = 4 is y = 4.

 

Sample Questions

Question 1. Convert Polar equation r = 10 sinθ to rectangular form

Solution:

Given r = 4 sinθ

Multiply both LHS and RHS by r

r2 = 4r sinθ

We already know, r2 = x2 + y2

Replace r2 by x2 + y2

x2 + y2 = 4r sinθ

Now, as we know y = r sinθ

So, replace r sinθ by y

x2 + y2 = 4 y

x2 + y2 – 4y = 0

Add 4 on both LHS and RHS

x2 + y2 + 4 – 4y  = 4

x2 + (y – 2)2 = 4

We conclude that the rectangular form of r = 4 sin theta is x2 + (y – 2)2 = 4

Question 2. Convert Polar equation r = 6 sinθ to rectangular form

Solution:

Given r = 6 sinθ

Multiply both LHS and RHS by r

r2 = 6r sinθ

We already know, r2 = x2 + y2

Replace r2 by x2 + y2

x2 + y2 = 6r sinθ

Now, as we know y = r sinθ

So, replace r sinθ by y

x2 + y2 = 6y

x2 + y2 – 6y = 0

Add 9 on both LHS and RHS

x2 + y2 + 9 – 6y  = 9

x2 + (y – 3)2 = 9

We conclude that the rectangular form of r = 6 sin theta is x2 + (y – 2)2 = 9

Question 3. Convert Polar equation r = 4 cosθ to rectangular form

Solution:

Given r = 4 cos theta

Multiply both LHS and RHS by r

r2 = 4 r cosθ

We already know, r2 = x2 + y2

Replace r2 by x2 + y2

x2 + y2 = 4r cosθ

Now, as we know x = r cosθ

So, replace r cosθ by y

x2 + y2 = 4 x

x2 + y2 – 4x = 0

Add 4 on both LHS and RHS

x2 + y2 + 4 – 4x = 4

(x – 2)2 + y2 = 4

We conclude that the rectangular form of r = 4 cos theta is (x – 2)2 + y2 = 4

Question 4. Given equation r = 6 cosθ, now convert it into a rectangular form

Solution:

Given r = 6 cosθ

Multiply both LHS and RHS by r

r2 = 6 r cosθ

We already know, r2 = x2 + y2

Replace r2 by x2 + y2

x2 + y2 = 6r cosθ

Now, as we know x = r cosθ

So, replace r cosθ by y

x2 + y2 = 6 x

x2 + y2 – 6x = 0

Add 9 on both LHS and RHS

x2 + y2 + 9 – 6x = 9

(x – 3)2 + y2 = 9

We conclude that the rectangular form of r = 6 cos theta is (x – 3)2 + y2 = 9

Question 5. Convert Polar equation r = 10 sinθ to rectangular form

Solution:

Given r = 10 sinθ

Multiply both LHS and RHS by r

r2 = 10r sinθ

We already know, r2 = x2 + y2

Replace r2 by x2 + y2

x2 + y2 = 10r sinθ

Now, as we know y = r sinθ

So, replace r sinθ by y

x2 + y2  = 10 y

x2 + y2 – 10y = 0

Add 25 on both LHS and RHS

x2 + y2 + 25 – 10y = 25

x2 + (y – 5)2 = 25

We conclude that the rectangular form of r = 10 sin theta is x2 + (y – 5)2 = 25



Last Updated : 31 Jan, 2024
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