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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*cos3(θ)−3*cos(θ)

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

  5. Substitute cos2(θ)=1−sin2(θ) to form a quadratic equation in terms of sin(θ) 4*sin2(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−sin2(54)) Alternatively, use the identity cos(54)=√(,10−2√(,5))/4

Final Answer

cos(54)=√(,10−2√(,5))/4

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