To convert the polar equation r = 5 cos(8θ) into a Cartesian equation, we start by recalling the relationships between polar and Cartesian coordinates:
- x = r cos(θ)
- y = r sin(θ)
- r = √(x² + y²)
- tan(θ) = y/x
From the polar equation, we know:
- r = 5 cos(8θ)
Now, we can express cos(8θ) using the identity:
- cos(8θ) = cos(θ)^{8} – 28 cos(θ)^{6}sin(θ)^{2} + 56 cos(θ)^{4}sin(θ)^{4} – 7 cos(θ)^{2}sin(θ)^{6}
However, it’s simpler to multiply both sides of r = 5 cos(8θ) by r:
- r² = 5r cos(8θ)
Substituting the relationships mentioned earlier, we have:
- r² = x² + y²
- r cos(8θ) = x
Now substituting into our equation:
- x² + y² = 5x
This can be rearranged to fit a standard form:
- x² – 5x + y² = 0
Completing the square for the x terms allows us to further understand this equation:
- (x – 2.5)² + y² = 6.25
This represents a circle with a center at (2.5, 0) and a radius of 2.5.
In conclusion, the Cartesian equation for the curve given by r = 5 cos(8θ) is
(x – 2.5)² + y² = 6.25, and it describes a circle.