Question

设 $\alpha$ 是 $3\text{-}4\text{-}5$ 直角三角形中最小角的弧度数。设 $\beta$ 是 $7\text{-}24\text{-}25$ 直角三角形中最小角的弧度数。用 $\alpha$ 表示， $\beta$ 等于什么？

Let $\alpha$ be the radian measure of the smallest angle in a $3\text{-}4\text{-}5$ right triangle. Let $\beta$ be the radian measure of the smallest angle in a $7\text{-}24\text{-}25$ right triangle. In terms of $\alpha,$ what is $\beta?$

$\dfrac{\alpha}{3}$

$\alpha-\dfrac{\pi}{8}$

$\dfrac{\pi}{2}-2\alpha$

$\dfrac{\alpha}{2}$

$\pi-4\alpha$

答案：C

知识点：三角恒等式直角三角形

难度评级：1570

解答：

$3\text{-}4\text{-}5$ 三角形的最小角满足 $\tan\alpha=\tfrac34$ 。因此 $7\text{-}24\text{-}25$ 三角形的最小角满足 $\tan\beta=\tfrac{7}{24}=\cot2\alpha$ $=\tan\!\left(\tfrac{\pi}{2}-2\alpha\right)$ 。所以 $\beta=\tfrac{\pi}{2}-2\alpha$ 。因此正确答案是 C。 $\begin{aligned} &\tan2\alpha=\frac{2\cdot\frac34}{1-\frac{9}{16}} \\ &=\frac{3/2}{7/16}=\frac{24}{7}. \end{aligned}$

The smallest angle of the $3\text{-}4\text{-}5$ triangle has $\tan\alpha=\tfrac34.$ Then $\begin{aligned} &\tan2\alpha=\frac{2\cdot\frac34}{1-\frac{9}{16}} \\ &=\frac{3/2}{7/16}=\frac{24}{7}. \end{aligned}$ The smallest angle of the $7\text{-}24\text{-}25$ triangle has $\tan\beta=\tfrac{7}{24}=\cot2\alpha$ $=\tan\!\left(\tfrac{\pi}{2}-2\alpha\right).$ Hence $\beta=\tfrac{\pi}{2}-2\alpha.$ Thus, the correct answer is C.

2024 AMC 12A 第 10 题

答案：C

难度评级：1570

解答：

其他年份的第 10 题