Evaluate arcsin(sin((6pi)/7))

Problem

arcsin(sin((6*π)/7))

Solution

1. Identify the range of the principal inverse sine function. The output of arcsin(x) must fall within the interval [−π/2,π/2]

2. Check if the input angle (6*π)/7 is within the principal range. Since (6*π)/7≈0.857*π it is greater than π/2 (which is 0.5*π, so the identity arcsin(sin(θ))=θ cannot be applied directly.

3. Apply the sine supplement identity sin(θ)=sin(π−θ) to find an equivalent angle within the principal range.

4. Calculate the reference angle.

π−(6*π)/7=(7*π)/7−(6*π)/7

π−(6*π)/7=π/7

5. Verify that the new angle π/7 is within the interval [−π/2,π/2] Since 0<1/7<1/2 the condition is satisfied.

6. Evaluate the expression using the property arcsin(sin(π/7))=π/7

Final Answer

arcsin(sin((6*π)/7))=π/7

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