Final answer:
The slope of the tangent line to the curve r = sin(2θ) at θ = π/4 is -1. This value was determined by taking the derivative of the polar function r with respect to θ and evaluating it along with the original function r at θ = π/4.
Step-by-step explanation:
To find the slope of the tangent line to the curve r = sin(2θ) at θ = π/4, we first need to calculate the derivative of the polar function with respect to θ. This derivative, denoted as dr/dθ, will give us the rate at which r changes with θ. To find the actual slope, we use the formula dy/dx = (dr/dθ*sin(θ) + r*cos(θ))/(dr/dθ*cos(θ) - r*sin(θ)).
First, let's find dr/dθ: dr/dθ = d(sin(2θ))/dθ = 2*cos(2θ). At θ = π/4, 2*cos(2* π/4) = 2*cos(π/2) = 0.
Next, evaluate r at θ = π/4: r = sin(2*u03c0/4) = sin(π/2) = 1.
Now plug these values into the slope formula: dy/dx = (0*sin(π/4) + 1*cos(π/4))/(0*cos(π/4) - 1*sin(π/4)) = (0 + √2/2)/(0 - √2/2) = -1.
Therefore, the slope of the tangent line to the curve at θ = π/4 is -1.