Find the Exact Value cos(pi/5)

Problem

cos(π/5)

Solution

1. Identify the angle in degrees to recognize its properties. Since π radians is 180^∘ the angle is 360^∘/10=36^∘

2. Use the multiple angle identity for sine, specifically 2*θ=36^∘ and 3*θ=54^∘ noting that sin(2*θ)=cos(3*θ) because 2*θ+3*θ=90^∘

3. Apply the double and triple angle formulas to the equation sin(2*θ)=cos(3*θ) where θ=18^∘

2*sin(θ)*cos(θ)=4*cos^3(θ)−3*cos(θ)

4. Divide by cos(θ) (since cos(18^∘)≠0 and rearrange into a quadratic equation in terms of sin(θ)

2*sin(θ)=4*cos^2(θ)−3

2*sin(θ)=4*(1−sin^2(θ))−3

4*sin^2(θ)+2*sin(θ)−1=0

5. Solve the quadratic equation using the quadratic formula x=(−b±√(,b^2−4*a*c))/(2*a)

sin(18^∘)=(−2+√(,4−4*(4)*(−1)))/8

sin(18^∘)=(−1+√(,5))/4

6. Use the half-angle identity or the double-angle identity for cosine to find cos(36^∘)

cos(2*θ)=1−2*sin^2(θ)

cos(36^∘)=1−2*((−1+√(,5))/4)^2

7. Simplify the expression to find the exact value.

cos(36^∘)=1−2*((1−2√(,5)+5)/16)

cos(36^∘)=1−(6−2√(,5))/8

cos(36^∘)=(8−6+2√(,5))/8

cos(36^∘)=(2+2√(,5))/8

cos(36^∘)=(1+√(,5))/4

Final Answer

cos(π/5)=(1+√(,5))/4

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