Find the Exact Value sin(arctan(-1))

Problem

sin(arctan(−1))

Solution

1. Identify the inner function value. Let θ=arctan(−1)

2. Determine the range of the arctangent function. The range of arctan(x) is (−π/2,π/2)

3. Solve for θ such that tan(θ)=−1 within the specified range.

4. Find the angle. Since tan(−π/4)=−1 we have θ=−π/4

5. Substitute the angle back into the outer sine function to evaluate sin(−π/4)

6. Apply the unit circle value for sin(−π/4) which is −√(,2)/2

Final Answer

sin(arctan(−1))=−√(,2)/2

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