Find the Exact Value sin(2arccos(1/2))

Problem

sin(2*arccos(1/2))

Solution

1. Identify the inner value of the inverse trigonometric function. We know that arccos(1/2) represents the angle θ in the interval [0,π] such that cos(θ)=1/2

2. Evaluate the inverse cosine. Since cos(π/3)=1/2 we have:

θ=arccos(1/2)=π/3

3. Substitute the angle back into the original expression:

sin(2⋅π/3)=sin((2*π)/3)

4. Apply the sine value for the angle in the second quadrant. The reference angle for (2*π)/3 is π/3 and sine is positive in the second quadrant:

sin((2*π)/3)=√(,3)/2

Final Answer

sin(2*arccos(1/2))=√(,3)/2

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