To find the slope of the tangent line to a polar curve at a specific point, we can use the formula for the slope in polar coordinates:
m = (dr/dθ) / (r sin(θ) + r’ cos(θ))
Where:
- r is the radius as a function of θ (in this case, r = 2 sin(θ)),
- dr/dθ is the derivative of r with respect to θ,
- r’ is the radius r evaluated at θ.
First, we need to compute the derivative of r:
r = 2 sin(θ)
dr/dθ = 2 cos(θ)
Next, we’ll evaluate r and dr/dθ at θ = π/3:
- r(π/3) = 2 sin(π/3) = 2 * (√3/2) = √3
- dr/dθ(π/3) = 2 cos(π/3) = 2 * (1/2) = 1
Now, we can substitute r and dr/dθ back into the slope formula:
m = (1) / (√3 sin(π/3) + 1 * cos(π/3))
Calculating the sine and cosine values:
- sin(π/3) = √3/2
- cos(π/3) = 1/2
Substituting these values gives:
m = 1 / (√3 * (√3/2) + 1 * (1/2))
m = 1 / (√3 * √3/2 + 1/2)
m = 1 / (3/2 + 1/2) = 1 / 2
Thus, the slope of the tangent line to the polar curve r = 2 sin(θ) at the point where θ = π/3 is 1/2.