Convert the polar equation to rectangular form.r = 81 − cos 𝜃

Sure, here are the steps to convert the polar equation r = 81 - cos θ to rectangular form:

1. Recall the relationships between polar and rectangular coordinates: x = rcosθ and y = rsinθ.

2. Substitute r = 81 - cos θ into the equations for x and y. This gives us x = (81 - cos θ)cos θ and y = (81 - cos θ)sin θ.

3. Now, we need to eliminate θ from these equations. We can do this by using the Pythagorean identity sin²θ + cos²θ = 1.

4. Rearrange the identity to get cos²θ = 1 - sin²θ.

5. Substitute this into the equation for x to get x = (81 - (1 - sin²θ))cos θ = 80cos θ + sin²θ cos θ.

6. Now, we can substitute y/rsinθ for sinθ in the equation for x to get x = 80cos θ + (y²/r²)cos θ.

7. Finally, we can substitute x/rcosθ for cosθ to get x = 80(x/r) + (y²/r²)(x/r), which simplifies to x² = 80x + y².

So, the rectangular form of the polar equation r = 81 - cos θ is x² = 80x + y².