1979 AMC 12 第 20 题

Question

若

a=\frac12

且

(a+1)(b+1)=2

，则

\operatorname{Arctan}a+\operatorname{Arctan}b

的弧度数等于

If

a=\frac12

and

(a+1)(b+1)=2,

then the radian measure of

\operatorname{Arctan}a+\operatorname{Arctan}b

equals

\frac\pi2

\frac\pi3

\frac\pi4

\frac\pi5

\frac\pi6

先由所给乘积等式解出

b

First solve the given product equation for

b

对两个正的反正切角使用正切加法公式

Use the tangent addition formula on the two positive inverse-tangent angles

代入

a=\frac{1}{2}

，得到

b=\frac{1}{3}

。令

u=\operatorname{Arctan}a

，

v=\operatorname{Arctan}b

，则

\begin{aligned} \tan(u+v)&=\frac{a+b}{1-ab}\\ &=\frac{\frac{1}{2}+\frac{1}{3}}{1-\frac{1}{6}}\\ &=1 \end{aligned}\text{。}

两个角都为正，且它们的和小于

\frac{\pi}{2}

，所以

u+v=\frac{\pi}{4}

。

因此，正确答案是 C。

Substituting

a=\frac{1}{2}

gives

b=\frac{1}{3}.

If

u=\operatorname{Arctan}a

and

v=\operatorname{Arctan}b,

then

\begin{aligned} \tan(u+v)&=\frac{a+b}{1-ab}\\ &=\frac{\frac{1}{2}+\frac{1}{3}}{1-\frac{1}{6}}\\ &=1. \end{aligned}

Both angles are positive and their sum is less than

\frac{\pi}{2},

so

u+v=\frac{\pi}{4}.

Therefore, the correct answer is C.

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