Find the Exact Value sin(54)

Problem

sin(54^∘)

Solution

1. Identify the angle in terms of a known relationship. Let θ=18^∘ Then 54^∘=3*θ We also know that 2*θ=36^∘ and 3*θ=54^∘ which are complementary because 2*θ+3*θ=90^∘

2. Set up the equation using the complementary angle identity sin(3*θ)=cos(2*θ)

3. Apply the triple-angle formula for sine and the double-angle formula for cosine:

3*sin(θ)−4*sin^3(θ)=1−2*sin^2(θ)

4. Rearrange the equation into a polynomial form by letting x=sin(θ)

4*x^3−2*x^2−3*x+1=0

5. Factor the polynomial. Since x=1 (which corresponds to sin(90^∘) is a root, we divide by (x−1)

(x−1)*(4*x^2+2*x−1)=0

6. Solve the quadratic part 4*x^2+2*x−1=0 using the quadratic formula:

x=(−2±√(,2^2−4*(4)*(−1)))/(2*(4))

x=(−2±√(,20))/8

x=(−1±√(,5))/4

7. Select the positive root for sin(18^∘) since 18^∘ is in the first quadrant:

sin(18^∘)=(√(,5)−1)/4

8. Use the double-angle identity cos(2*θ)=1−2*sin^2(θ) to find cos(36^∘) which is equal to sin(54^∘)

sin(54^∘)=1−2*((√(,5)−1)/4)^2

9. Simplify the expression:

sin(54^∘)=1−2*((5−2√(,5)+1)/16)

sin(54^∘)=1−(6−2√(,5))/8

sin(54^∘)=(8−6+2√(,5))/8

sin(54^∘)=(2+2√(,5))/8

sin(54^∘)=(1+√(,5))/4

Final Answer

sin(54^∘)=(1+√(,5))/4

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