tex]: Ответ должен получиться 1,732...

July 2022 1 7 Report

Упростите выражение[tex] \frac{tg \frac{ \pi }{15}+tg \frac{4 \pi }{15} }{1-tg \frac{4 \pi }{15}*tg \frac{ \pi }{15} } [/tex]:

Ответ должен получиться 1,732

Answers & Comments

Apofeoz $\frac{tg\frac{\pi}{15}+tg\frac{4\pi}{15}}{1-tg\frac{4\pi}{15}\bullet tg\frac{\pi}{15}}=\frac{\frac{\sin(\frac{\pi+4\pi}{15})}{\cos\frac{\pi}{15}\cos\frac{4 \pi }{15}}}{1-\frac{\sin\frac{4\pi}{15}}{\cos\frac{4\pi}{15}}\bullet \frac{\sin\frac{\pi}{15}}{\cos\frac{\pi}{15}}}=\frac{\sin\frac{\pi}{3}}{\cos\frac{\pi}{15}\cos\frac{4 \pi }{15}(1-\frac{\sin\frac{4\pi}{15}\sin\frac{\pi}{15}}{\cos\frac{4\pi}{15}\cos\frac{\pi}{15}})}=$

$=\frac{\frac{\sqrt3}{2}}{\cos\frac{\pi}{15}\cos\frac{4 \pi 
}{15}-\sin\frac{4\pi}{15}\sin\frac{\pi}{15}}=\frac{\sqrt3}{2(\frac{1}{2}(\cos\frac{-3\pi}{15}+\cos\frac{5\pi}{15})-\frac{1}{2}(\\cos\frac{-3\pi}{15}-\cos\frac{5\pi}{15}))}=\\\\=\frac{\sqrt3}{\cos\frac{-3\pi}{15}+\cos\frac{5\pi}{15}-\cos\frac{-3\pi}{15}+\cos\frac{5\pi}{15}}=\frac{\sqrt3}{\cos\frac{\pi}{5}+\cos\frac{\pi}{3}-\cos\frac{\pi}{5}+\cos\frac{\pi}{3}}=\frac{\sqrt3}{\frac{1}{2}+\frac{1}{2}}=\frac{\sqrt3}{1}=\\\\=\sqrt3.\\\\\sqrt3\approx1.732$

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