Simplify tan(arcsin(-( square root of 3)/2))

Problem

tan(arcsin(−√(,3)/2))

Solution

1. Evaluate the inner inverse trigonometric function arcsin(−√(,3)/2)

2. Identify the range of the arcsine function, which is [−π/2,π/2]

3. Determine the angle θ such that sin(θ)=−√(,3)/2 within that range.

θ=−π/3

4. Substitute the angle back into the tangent function.

tan(−π/3)

5. Calculate the value of the tangent function using the identity tan(θ)=sin(θ)/cos(θ)

tan(−π/3)=(−√(,3)/2)/1/2

6. Simplify the resulting fraction.

tan(−π/3)=−√(,3)

Final Answer

tan(arcsin(−√(,3)/2))=−√(,3)

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