Find the Exact Value sin(pi/5)

Identify the angle as 36 and use the multiple angle relationship 2*θ=π−3*θ where θ=π/5

Apply the sine and cosine double and triple angle identities to the equation sin(2*θ)=sin(3*θ)

Simplify the resulting equation 2*sin(θ)*cos(θ)=3*sin(θ)−4*sin3(θ) by dividing by sin(θ) assuming sin(θ)≠0

Substitute sin2(θ)=1−cos2(θ) to get a quadratic equation in terms of cos(θ) which is 4*cos2(θ)−2*cos(θ)−1=0

Solve for cos(θ) using the quadratic formula, selecting the positive root cos(π/5)=(1+√(,5))/4 because the angle is in the first quadrant.

Calculate sin(π/5) using the Pythagorean identity sin(θ)=√(,1−cos2(θ))

Evaluate the expression √(,1−((1+√(,5))/4)2)=√(,1−(6+2√(,5))/16)

Simplify the fraction to reach the final exact value.