Find the Exact Value sin(40 degrees )

Problem

sin(40^∘)

Solution

1. Identify the nature of the angle 40^∘ Since 40^∘ is not a standard reference angle (like 30^∘ 45^∘ or 60^∘, its exact value cannot be expressed using simple square roots or basic fractions.

2. Relate the angle to the triple-angle formula for sine, which is sin(3*θ)=3*sin(θ)−4*sin^3(θ) If we let θ=40^∘ then 3*θ=120^∘

3. Substitute the known value sin(120^∘)=√(,3)/2 into the formula to create a cubic equation.

√(,3)/2=3*sin(40^∘)−4*sin^3(40^∘)

4. Recognize that solving this cubic equation for sin(40^∘) involves complex numbers or nested radicals (Casus irreducibilis), which does not simplify to a standard "exact value" form typically expected in trigonometry unless expressed in terms of roots of a cubic polynomial.

5. Conclude that the most precise way to write the exact value without using cubic roots of complex numbers is simply the expression itself, as it is a transcendental-like algebraic number.

Final Answer

sin(40^∘)=sin(40^∘)

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