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

Problem

cos(arcsin(−√(,3)/2))

Solution

1. Identify the inner value y=arcsin(−√(,3)/2) This represents the angle in the interval [−π/2,π/2] whose sine is −√(,3)/2

2. Evaluate the inverse sine function. Since sin(π/3)=√(,3)/2 and the sine function is odd, sin(−π/3)=−√(,3)/2

3. Substitute the angle back into the outer cosine function. We now need to find cos(−π/3)

4. Apply the property that cosine is an even function, meaning cos(−θ)=cos(θ)

5. Simplify the expression using the known value cos(π/3)=1/2

Final Answer

cos(arcsin(−√(,3)/2))=1/2

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