Find the Exact Value cos(40)

Problem

cos(40^∘)

Solution

1. Identify the nature of the angle. The angle 40^∘ is not one of the standard reference angles (0^∘,30^∘,45^∘,60^∘,90^∘ for which exact values are typically expressed using simple radicals.

2. Determine if an exact algebraic form exists. While cos(40^∘) can be expressed using the cubic formula (since it relates to cos(120^∘) via the triple angle identity cos(3*θ)=4*cos^3(θ)−3*cos(θ), such expressions involve nested radicals and complex numbers (casus irreducibilis).

3. State the standard mathematical convention. In most contexts, unless specifically asked for the roots of a cubic equation, the exact value is simply written as the expression itself.

Final Answer

cos(40^∘)=cos(40^∘)

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