Find the Exact Value arcsin(sin((5pi)/8))

Problem

arcsin(sin((5*π)/8))

Solution

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

2. Determine if the input angle (5*π)/8 lies within this range. Since (5*π)/8>(4*π)/8 (which is π/2, the angle is outside the principal range.

3. Apply the identity sin(θ)=sin(π−θ) to find a reference angle within the required interval.

4. Calculate the equivalent angle:

π−(5*π)/8=(8*π)/8−(5*π)/8

(8*π−5*π)/8=(3*π)/8

5. Verify that (3*π)/8 is within the interval [−π/2,π/2] Since 0<3/8<1/2 the condition is met.

6. Evaluate the expression using the property arcsin(sin(x))=x for x∈[−π/2,π/2]

Final Answer

arcsin(sin((5*π)/8))=(3*π)/8

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