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*cos3(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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