Find the Exact Value cos(54)

Problem

cos(54^∘)

Solution

1. Identify the angle as a multiple of 18^∘ Let θ=18^∘ We are looking for cos(3*θ)

2. Use the relationship 2*θ=90^∘−3*θ which implies sin(2*θ)=cos(3*θ)

3. Expand both sides using double-angle and triple-angle identities: 2*sin(θ)*cos(θ)=4*cos^3(θ)−3*cos(θ)

4. Divide by cos(θ) (since cos(18^∘)≠0 to get 2*sin(θ)=4*cos^2(θ)−3

5. Substitute cos^2(θ)=1−sin^2(θ) to form a quadratic equation in terms of sin(θ) 4*sin^2(θ)+2*sin(θ)−1=0

6. Solve the quadratic equation using the quadratic formula to find sin(18^∘)=(−1+√(,5))/4

7. Apply the identity cos(54^∘)=sin(36^∘) because they are complementary angles.

8. Calculate sin(36^∘) using the double-angle formula sin(36^∘)=2*sin(18^∘)*cos(18^∘) or the identity cos(54^∘)=√(,1−sin^2(54^∘)) Alternatively, use the identity cos(54^∘)=√(,10−2√(,5))/4

Final Answer

cos(54^∘)=√(,10−2√(,5))/4

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