Find the Exact Value sin(18)

Problem

sin(18)

Solution

1. Set up the relationship by letting θ=18 which implies 5*θ=90

2. Split the equation into 2*θ+3*θ=90 which can be rewritten as 2*θ=90−3*θ

3. Apply the sine function to both sides to get sin(2*θ)=sin(90−3*θ)

4. Use the co-function identity to rewrite the right side as sin(2*θ)=cos(3*θ)

5. Substitute the double-angle and triple-angle identities: 2*sin(θ)*cos(θ)=4*cos3(θ)−3*cos(θ)

6. Divide both sides by cos(θ) (since cos(18)≠0 to obtain 2*sin(θ)=4*cos2(θ)−3

7. Replace cos2(θ) with 1−sin2(θ) to get 2*sin(θ)=4*(1−sin2(θ))−3

8. Rearrange the equation into a quadratic form: 4*sin2(θ)+2*sin(θ)−1=0

9. Solve using the quadratic formula x=(−b±√(,b2−4*a*c))/(2*a) where x=sin(θ)

10. Calculate the roots: sin(θ)=(−2±√(,4−4*(4)*(−1)))/8=(−2±√(,20))/8=(−2±2√(,5))/8

11. Select the positive root (−1+√(,5))/4 because 18 is in the first quadrant where sine is positive.

Final Answer

sin(18)=(√(,5)−1)/4

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