How do you find the slope of the tangent line to the polar curve r = 2 sin(θ) at the point specified by θ = π/3?

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To find the slope of the tangent line to a polar curve given by the equation r(θ) = 2 sin(θ) at the specific point where θ = π/3, we follow these steps:

1. Calculate r at θ = π/3

Substituting θ = π/3 into the equation, we get:

r(π/3) = 2 sin(π/3) = 2 * (√3/2) = √3

This means that at θ = π/3, the point in polar coordinates is (√3, π/3).

2. Use the formulas for conversion to Cartesian coordinates

The conversion from polar to Cartesian coordinates is given by:

x = r cos(θ)

y = r sin(θ)

Substituting our values, we find:

x = √3 cos(π/3) = √3 * (1/2) = √3/2
y = √3 sin(π/3) = √3 * (√3/2) = 3/2

3. Determine the derivatives for the slope

The slope of the tangent line in polar coordinates can be found using the formula:

dy/dx = (dr/dθ) / (dθ/dθ)

Now, we need to compute dr/dθ. Let’s differentiate r = 2 sin(θ):

dr/dθ = 2 cos(θ)

Next, since dθ/dθ = 1, we have:

dy/dx = 2 cos(θ)

4. Evaluate the derivative at θ = π/3

We need to evaluate dy/dx at θ = π/3:

dy/dx = 2 cos(π/3) = 2 * (−1/2) = −1

5. Conclusion

The slope of the tangent line to the polar curve r = 2 sin(θ) at the point (√3, π/3) is -1.

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