Find the Exact Value arcsin(sin((10pi)/7))

Problem

arcsin(sin((10*π)/7))

Solution

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

2. Determine if the input angle (10*π)/7 is within the range. Since (10*π)/7>π/2 we must find a coterminal or related angle within the principal range that has the same sine value.

3. Find a coterminal angle by subtracting 2*π from the original angle.

(10*π)/7−2*π=(10*π)/7−(14*π)/7

(10*π)/7−2*π=−(4*π)/7

4. Apply the property sin(θ)=sin(π−θ) to find an angle within the required interval.

π−(−(4*π)/7)=(7*π)/7+(4*π)/7

π−(−(4*π)/7)=(11*π)/7

This is still outside the range.

5. Use the property sin(θ)=−sin(θ−π) or observe the quadrant. The angle (10*π)/7 is in the third quadrant. The reference angle is:

(10*π)/7−π=(3*π)/7

6. Determine the value in the range [−π/2,π/2] that has the same sine value. Since sin((10*π)/7) is negative and the reference angle is (3*π)/7 the corresponding angle in the principal range is −(3*π)/7

−π/2≤−(3*π)/7≤π/2

Final Answer

arcsin(sin((10*π)/7))=−(3*π)/7

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