To find the exact value of arctan(sin(π/2)), we first need to calculate sin(π/2).
We know that sin(π/2) is equal to 1. This is because at an angle of π/2 radians (or 90 degrees), the sine function reaches its maximum value of 1.
Now we substitute this value into our original expression:
arctan(sin(π/2)) = arctan(1)
The next step is to determine arctan(1). The arctangent function gives us the angle whose tangent is the value we input. The tangent of what angle equals 1? The answer is π/4 radians (or 45 degrees), because:
tan(π/4) = 1
Thus, we can conclude:
arctan(sin(π/2)) = arctan(1) = π/4
In summary, the exact value of arctan(sin(π/2)) is π/4.