Find the Exact Value arcsin(sin((6pi)/7))

Problem

arcsin(sin((6*π)/7))

Solution

1. Identify the range of the principal inverse sine function, which is [−π/2,π/2]

2. Determine if the input angle (6*π)/7 lies within this range. Since (6*π)/7>π/2 the property arcsin(sin(θ))=θ cannot be applied directly.

3. Apply the sine identity sin(θ)=sin(π−θ) to find a reference angle within the principal range.

4. Calculate the equivalent angle:

π−(6*π)/7=(7*π)/7−(6*π)/7

(7*π)/7−(6*π)/7=π/7

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

6. Evaluate the expression using the equivalent angle:

arcsin(sin(π/7))=π/7

Final Answer

arcsin(sin((6*π)/7))=π/7

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