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

Problem

sec(arcsin(−√(,3)/2))

Solution

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

2. Recall that the range of the arcsine function is [−π/2,π/2]

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

θ=−π/3

4. Substitute the angle back into the original expression to find sec(−π/3)

sec(−π/3)

5. Apply the reciprocal identity sec(x)=1/cos(x)

sec(−π/3)=1/cos(−π/3)

6. Use the even function property cos(−x)=cos(x) to simplify the denominator.

cos(−π/3)=cos(π/3)=1/2

7. Calculate the final value by taking the reciprocal.

1/(1/2)=2

Final Answer

sec(arcsin(−√(,3)/2))=2

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