Find the Exact Value tan(arcsin(( square root of 3)/2))

Problem

tan(arcsin(√(,3)/2))

Solution

1. Identify the inner value arcsin(√(,3)/2) as an angle θ such that sin(θ)=√(,3)/2 within the interval [−π/2,π/2]

2. Determine the angle θ from the unit circle where the sine value is √(,3)/2

θ=π/3

3. Substitute the angle back into the original expression to find tan(θ)

tan(π/3)

4. Evaluate the tangent of π/3 using the ratio of sine to cosine.

tan(π/3)=sin(π/3)/cos(π/3)

tan(π/3)=√(,3)/2/1/2

5. Simplify the fraction to find the final exact value.

tan(π/3)=√(,3)

Final Answer

tan(arcsin(√(,3)/2))=√(,3)

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