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

Problem

sec(arcsin(1/2))

Solution

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

2. Determine the value of θ using the unit circle or special triangles, noting that sin(30)=1/2

θ=π/6

3. Substitute the angle back into the outer function to find sec(π/6)

sec(π/6)=1/cos(π/6)

4. Evaluate the cosine of the angle, where cos(π/6)=√(,3)/2

sec(π/6)=1/√(,3)/2

5. Simplify the fraction by taking the reciprocal.

sec(π/6)=2/√(,3)

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

2/√(,3)⋅√(,3)/√(,3)=(2√(,3))/3

Final Answer

sec(arcsin(1/2))=(2√(,3))/3

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