Find the Exact Value arcsin(sin((11pi)/4))

Problem

arcsin(sin((11*π)/4))

Solution

1. Simplify the angle inside the sine function by finding its coterminal equivalent within the standard interval [0,2*π)

(11*π)/4=(8*π+3*π)/4

(11*π)/4=2*π+(3*π)/4

2. Evaluate the sine of the simplified angle, noting that sin(2*π+θ)=sin(θ)

sin((11*π)/4)=sin((3*π)/4)

sin((3*π)/4)=√(,2)/2

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

arcsin(√(,2)/2)=θ

sin(θ)=√(,2)/2

4. Determine the value of θ within the interval [−π/2,π/2] that satisfies the equation.

θ=π/4

Final Answer

arcsin(sin((11*π)/4))=π/4

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