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Problem 825

AMC 10/12, early questions

Geometry Difficulty 3.7 Find the answer China Mathematical Competition · China

In $\triangle ABC$ , $\tan A$ and $\tan B$ are the two roots of equation $x^2 - 10x + 6 = 0$ . Then the value of $\cos C$ is ______.

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Official solution

By the condition, we know that $\tan A + \tan B = 10$ , $\tan A \tan B = 6$ . Thus, $\begin{aligned} \tan C &= \tan(\pi - A - B) \\ &= -\tan(A + B) \\ &= -\frac{\tan A + \tan B}{1 - \tan A \tan B} = 2. \end{aligned}$ Therefore, $C$ is an acute angle, and thus $\cos C = \frac{1}{\sqrt{1 + \tan^2 C}} = \frac{1}{\sqrt{5}} = \frac{\sqrt{5}}{5}$ .

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