Simplify sin(arccos(-( square root of 3)/2))

Problem

sin(arccos(−√(,3)/2))

Solution

1. Identify the inner value x=−√(,3)/2 and the range of the inverse cosine function, which is [0,π]

2. Evaluate the inner expression θ=arccos(−√(,3)/2) by finding the angle in the second quadrant where the cosine is −√(,3)/2

θ=(5*π)/6

3. Substitute the angle back into the outer sine function.

sin((5*π)/6)

4. Calculate the sine of (5*π)/6 using the unit circle, noting that the sine of an angle in the second quadrant is positive.

sin((5*π)/6)=1/2

Final Answer

sin(arccos(−√(,3)/2))=1/2

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