Problem 78 - Algebra and Trigonometry Real Mathematics, Real People

Question

Convert the polar equation to rectangular form. $$r=\frac{2}{1+\sin \theta}$$

Step-by-Step Solution

Answer

The rectangular form of the polar equation is \(x^4 + y^4 + 1 + x^2y^2 - y^3 + y^2 = x^2 + y^2 - y\).

Step 1: Substitution for sin θ

Express \(\sin{\theta}\) in terms of \'y\' and \'r\'. This would give \(\sin{\theta} = y/r\). Substitute this in the given polar equation, to get \(r = 2/(1+(y/r))\).

Step 2: Solve for r

Multiply through by r to eliminate the denominator, the equation becomes \(r^2 = 2r - 2y\).

Step 3: Substitution for r

Substitute for \'r\' as defined in terms of rectangular coordinates. Substitute \(r^2\) as \(x^2 + y^2\), the equation becomes \(x^2 + y^2 = 2\sqrt{x^2 + y^2} - 2y\)

Step 4: Simplify Equation

Rearrange and simplify equation to a more familiar rectangular form (preferably like standard form of ellipse, parabola, hyperbola, or circle). Here, it simplifies to \(x^2 + y^2 - 2y = 2\sqrt{x^2 + y^2}\). Square both sides to eliminate the root to get \(x^4 + 4x^2y^2 + y^4 - 4x^2y – 4y^3 + 4y^2 = 4x^2 + 4y^2-4y\). Finally, simplify by dividing whole equation by 4, which then results to \(x^4 + y^4 + 1 + x^2y^2 - y^3 + y^2 = x^2 + y^2 - y\).