Find the Exact Value tan(arcsin(1/2))

Problem

tan(arcsin(1/2))

Solution

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

2. Determine the angle θ from the unit circle where the sine value is 1/2

θ=π/6

3. Substitute the angle back into the original expression to find the tangent of that angle.

tan(π/6)

4. Evaluate the tangent function using the ratio of sine to cosine for the identified angle.

tan(π/6)=sin(π/6)/cos(π/6)

tan(π/6)=1/2/√(,3)/2

5. Simplify the fraction by multiplying by the reciprocal.

1/2⋅2/√(,3)=1/√(,3)

6. Rationalize the denominator by multiplying the numerator and denominator by √(,3)

1/√(,3)⋅√(,3)/√(,3)=√(,3)/3

Final Answer

tan(arcsin(1/2))=√(,3)/3

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