Question

一个指针钟从午夜开始走了 $2025$ 分钟后停止。停止时，时针和分针之间的锐角的正切值是多少？

An analog clock starts at midnight and runs for $2025$ minutes before stopping. What is the tangent of the acute angle between the hour hand and the minute hand when the clock stops?

$0$

$\sqrt{2} - 1$

$2 - \sqrt{2}$

$\dfrac{\sqrt{2}}{2}$

$3 - \sqrt{2}$

答案：B

知识点：时钟三角恒等式

难度评级：1840

解答：

$2025$ 分钟是 $33$ 小时 $45$ 分钟，按 $12$ 小时取余后为 $9{:}45$ 。分针指向 $270^\circ$ ，时针指向 $9.75 \times 30^\circ = 292.5^\circ$ ，所以两针之间的锐角是 $22.5^\circ$ 。由半角值， $\tan 22.5^\circ = \sqrt{2} - 1$ 。

所以正确答案是 B。

$2025$ minutes is $33$ hours $45$ minutes, which modulo $12$ hours reads $9{:}45.$ The minute hand points at $270^\circ$ and the hour hand at $9.75 \times 30^\circ = 292.5^\circ,$ so the acute angle between them is $22.5^\circ.$ Using the half-angle value, $\tan 22.5^\circ = \sqrt{2} - 1.$

Thus, the correct answer is B.

2025 AMC 12B 第 16 题

答案：B

难度评级：1840

解答：

其他年份的第 16 题