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Find the Exact Value cos(20)

Problem

cos(20)

Solution

  1. Identify the task as finding the exact trigonometric value of cos(20)

  2. Recognize that 20 is not a standard angle (like 30 45 or 60 for which a simple radical expression exists.

  3. Relate the angle to the triple angle formula for cosine, which is cos(3*θ)=4*cos3(θ)−3*cos(θ)

  4. Substitute θ=20 into the formula to get cos(60)=4*cos3(20)−3*cos(20)

  5. Evaluate cos(60)=1/2 resulting in the cubic equation 4*x3−3*x−1/2=0 where x=cos(20)

  6. Conclude that because this cubic equation does not have rational roots, the exact value is typically expressed using the roots of this polynomial or via complex numbers (De Moivre's Theorem), which simplifies back to the original expression.

Final Answer

cos(20)=cos(20)

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