examday

/TMUA

As is shown in the graph, the unit circle intersects with the $x$-axis at the point $A(1,0)$. The terminal sides of the acute angles $\alpha,\beta,\alpha-\beta$ are formed by starting from $O(0,0)$ and their initial sides along the positive $x$-axis.

Which of the following statements must be true?

• I. The area of the sector $A_1OP_1$ is $\dfrac12(\alpha-\beta)$.
• II. If $A_1$ and $P$ overlap, $|AP_1|=2\sin\beta$.
• III. If $\alpha=\dfrac\pi3$, the area of the quadrilateral $AA_1P_1O$ ranges from $\dfrac{\sqrt3}4$ to $1$.