Find the Exact Value sin(pi/10)

Problem

sin(π/10)

Solution

1. Identify the angle in degrees to recognize the problem. Since π radians is 180^∘ the angle is 18^∘ Let θ=18^∘

2. Set up a relationship between 2*θ and 3*θ Since 5*θ=90^∘ we have 2*θ=90^∘−3*θ

3. Apply the sine function to both sides.

sin(2*θ)=sin(90^∘−3*θ)

4. Use the cofunction identity sin(90^∘−x)=cos(x)

sin(2*θ)=cos(3*θ)

5. Expand using double-angle and triple-angle identities.

2*sin(θ)*cos(θ)=4*cos^3(θ)−3*cos(θ)

6. Divide by cos(θ) since cos(18^∘)≠0

2*sin(θ)=4*cos^2(θ)−3

7. Substitute the Pythagorean identity cos^2(θ)=1−sin^2(θ) to create a quadratic equation in terms of sin(θ)

2*sin(θ)=4*(1−sin^2(θ))−3

8. Rearrange the equation into standard quadratic form a*x^2+b*x+c=0

4*sin^2(θ)+2*sin(θ)−1=0

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

sin(θ)=(−2±√(,2^2−4*(4)*(−1)))/(2*(4))

10. Simplify the radical expression.

sin(θ)=(−2±√(,20))/8

sin(θ)=(−2±2√(,5))/8

sin(θ)=(−1±√(,5))/4

11. Select the positive root because 18^∘ is in the first quadrant, where sine is positive.

sin(π/10)=(√(,5)−1)/4

Final Answer

sin(π/10)=(√(,5)−1)/4

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