The expression is sin−1(cos53π). To simplify, convert cosine into sine.
Using the identity cosθ=sin(2π−θ) with θ=53π:
cos53π=sin(2π−53π)
To subtract the fractions, find a common denominator:
2π−53π
=105π−106π
=10−π
Therefore:
cos53π=sin(10−π)
Substituting back into the original expression:
sin−1(cos53π)
=sin−1(sin10−π)
For sin−1(sinx)=x, the value x must be in the range [−2π,2π].
Since −2π<10−π<2π, the value is in the valid range.
Therefore:
sin−1(sin10−π)=10−π
The answer is 10−π.