To find the exact value of arctan(sin(p/2)), we first need to evaluate sin(p/2).
The angle p/2 (or π/2 in radians) corresponds to 90 degrees. The sine of 90 degrees is one of the fundamental values in trigonometry, which is:
- sin(π/2) = 1
Now, substituting this back into our original expression:
- arctan(sin(π/2)) = arctan(1)
The arctan function, also known as the inverse tangent function, returns the angle whose tangent is the specified value. Therefore:
- arctan(1) is the angle where the tangent equals 1.
- This occurs at π/4 radians (or 45 degrees), since tan(π/4) = 1.
Thus, we have:
- arctan(sin(π/2)) = arctan(1) = π/4
In conclusion, the exact value of arctan(sin(π/2)) is π/4.