Find the Exact Value sin(40)

Problem

sin(40)

Solution

1. Identify the task as finding the exact trigonometric value of sin(40)

2. Determine if 40 is a standard angle on the unit circle. Standard angles are typically multiples of 30 or 45

3. Analyze the angle using the triple-angle formula sin(3*θ)=3*sin(θ)−4*sin3(θ) Setting θ=40 gives sin(120)=3*sin(40)−4*sin3(40)

4. Substitute the known value sin(120)=√(,3)/2 into the equation to get √(,3)/2=3*sin(40)−4*sin3(40)

5. Conclude that because this results in a cubic equation 8*x3−6*x+√(,3)=0 (where x=sin(40), the value cannot be expressed using only square roots and basic arithmetic. It is typically left in its trigonometric form or expressed using complex numbers via De Moivre's Theorem.

Final Answer

sin(40)=sin(40)

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