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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.
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.