Final answer:
The exact value of arccos(sin(pi/6)) is pi/3. This is calculated by first determining that sin(pi/6) equals 1/2 and then finding the angle whose cosine is 1/2 within the restricted domain of arccos, which is pi/3 or 60°.
Step-by-step explanation:
To find the exact value of
arccos(sin( π/6)), we can use the fact that sin(θ)=cos( π/2−θ) for all angles θ.
So, in this case, sin( π/6)=cos( π/2 - π/6). Now, simplify the angle:
π/2 - π/6 = 3π/6 - π/6 = 2π/6=π/3
Therefore,
arccos(sin( π/6))=arccos(cos( π/3)).
Now, recall that
arccos(cos(θ))=θ if 0≤θ≤π. In this case, π/3 is between 0 and π, so:
arccos(cos( π/3))= π/3
Therefore, the exact value of arccos(sin( π/6)) is π/3.