Maths Olympiad Prep

Library / /10 of 156

Geometry Difficulty 3.7 AMC 10/12 Find the answer China

In △ABC\triangle ABC, tan⁡A\tan A and tan⁡B\tan B are the two roots of equation x2−10x+6=0x^2 - 10x + 6 = 0. Then the value of cos⁡C\cos C is ______.

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

Solution

By the condition, we know that tan⁡A+tan⁡B=10\tan A + \tan B = 10, tan⁡Atan⁡B=6\tan A \tan B = 6. Thus,
tan⁡C=tan⁡(π−A−B)=−tan⁡(A+B)=−tan⁡A+tan⁡B1−tan⁡Atan⁡B=2. \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, CC is an acute angle, and thus cos⁡C=11+tan⁡2C=15=55\cos C = \frac{1}{\sqrt{1 + \tan^2 C}} = \frac{1}{\sqrt{5}} = \frac{\sqrt{5}}{5}.

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.