Find the Exact Value sin(pi/5)

Problem

sin(π/5)

Solution

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

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

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

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

5. 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.

6. Calculate sin(π/5) using the Pythagorean identity sin(θ)=√(,1−cos^2(θ))

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

8. Simplify the fraction to reach the final exact value.

Final Answer

sin(π/5)=√(,10−2√(,5))/4

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