Find a cartesian equation for the curve: r = 8 sin(θ) + 8 cos(θ).

Polar coordinates are used to represent different curves in the polar plane. Like cartesian coordinates, the polar coordinates have two parameters: the magnitude r and the phase angle θ. They are helpful in representing circular objects on the plane.

Answer: The cartesian equation for the curve r = 8 sin(θ) + 8 cos(θ) is (x − 4)² + (y − 4)² = 32.

Let's understand the solution in detail, step by step.

Explanation:

To convert a cartesian equation to a polar equation, we use the following substitutions:

x = r cos θ   ---- (1)

y = r sin θ    ---- (2)

Now, squaring and adding the adding the above equations, we get,

x² + y² = r² (cos²θ + sin²θ) = r²   (since, cos²θ + sin²θ = 1)

Hence, r = √(x² + y²)  ---- (3)

Now, we have r = 8 cos θ + 8 sin θ.

Now, multiply both sides by r, we get,

r² = 8 r cos θ + 8 r sin θ 

r² = 8 (r cos θ + r sin θ) 

From equation 1, equation 2 and equation 3, we get:

x² + y² = 8x + 8y 

This is the equation of a circle. We can represent the above equation in the standard form of circle.

x² − 8x + y² − 8y = 0 

Adding 32 on both the sides,

x² − 8x + 16 + y² − 8y + 16 = 16 + 16

Using the algebraic identity - (a - b)²:

(x − 4)² + (y − 4)² = 32

Therefore, we get the equation of a circle centered at the point (4, 4) with radius = √32 units.

Hence, the cartesian equation for the curve r = 8 sin(θ) + 8 cos(θ) is (x − 4)² + (y − 4)² = 32.