Вспомним одну тригонометрическую формулу:
![tg( \alpha + \beta )=\frac{tg \alpha + tg \beta }{1 - tg \alpha tg \beta}](https://tex.z-dn.net/?f=tg%28+%5Calpha+%2B+%5Cbeta+%29%3D%5Cfrac%7Btg+%5Calpha+%2B+tg+%5Cbeta+%7D%7B1+-+tg+%5Calpha+tg++%5Cbeta%7D)
Используя эту формулу, получаем:
![\frac{tg\frac{\pi}{10} + tg\frac{3\pi}{20}}{1-tg\frac{\pi}{10}tg\frac{3\pi}{20}}=tg(\frac{\pi}{10} + \frac{3\pi}{20})=tg\frac{5\pi}{20}=tg\frac{\pi}{4} = 1](https://tex.z-dn.net/?f=%5Cfrac%7Btg%5Cfrac%7B%5Cpi%7D%7B10%7D+%2B+tg%5Cfrac%7B3%5Cpi%7D%7B20%7D%7D%7B1-tg%5Cfrac%7B%5Cpi%7D%7B10%7Dtg%5Cfrac%7B3%5Cpi%7D%7B20%7D%7D%3Dtg%28%5Cfrac%7B%5Cpi%7D%7B10%7D+%2B+%5Cfrac%7B3%5Cpi%7D%7B20%7D%29%3Dtg%5Cfrac%7B5%5Cpi%7D%7B20%7D%3Dtg%5Cfrac%7B%5Cpi%7D%7B4%7D+%3D+1)
Ответ: 1
Узнаем стоимость магнитофона:
1) 8 000 - 100%
х - 10%
х=8 000·10:100=800 (р.)
8 000 -800=7200 (р.)
Выясним стоимость телевизора:
2) 7200 - 100%
х - 20%
х=7200·20:100=1440
7200 +1440 = 8640 (р.)
Ответ: 8640 рублей стоит телевизор.
По формуле синуса двойного аргумента:
![sin2x = \frac{ \sqrt{3} }{2}](https://tex.z-dn.net/?f=sin2x%20%3D%20%20%5Cfrac%7B%20%5Csqrt%7B3%7D%20%7D%7B2%7D%20)
![2x = (-1)^{n}\frac{ \pi }{3} + \pi n](https://tex.z-dn.net/?f=2x%20%3D%20%20%28-1%29%5E%7Bn%7D%5Cfrac%7B%20%5Cpi%20%7D%7B3%7D%20%2B%20%20%5Cpi%20n)
, n ∈ Z
![x = (-1)^{n}\frac{ \pi }{6} + \frac{\pi n}{2}](https://tex.z-dn.net/?f=x%20%3D%20%28-1%29%5E%7Bn%7D%5Cfrac%7B%20%5Cpi%20%7D%7B6%7D%20%2B%20%20%5Cfrac%7B%5Cpi%20n%7D%7B2%7D%20)
, n ∈ Z
![2cos( \frac{ \pi }{4}- \frac{x}{2} ) = \sqrt{3}](https://tex.z-dn.net/?f=2cos%28%20%5Cfrac%7B%20%5Cpi%20%7D%7B4%7D-%20%5Cfrac%7Bx%7D%7B2%7D%20%29%20%3D%20%5Csqrt%7B3%7D%20)
![cos( -\frac{ \pi }{4}+ \frac{x}{2} ) = \frac{ \sqrt{3} }{2}](https://tex.z-dn.net/?f=cos%28%20-%5Cfrac%7B%20%5Cpi%20%7D%7B4%7D%2B%20%5Cfrac%7Bx%7D%7B2%7D%20%29%20%3D%20%5Cfrac%7B%20%5Csqrt%7B3%7D%20%7D%7B2%7D%20)
![-\frac{ \pi }{4} + \frac{x}{2} =](https://tex.z-dn.net/?f=-%5Cfrac%7B%20%5Cpi%20%7D%7B4%7D%20%2B%20%5Cfrac%7Bx%7D%7B2%7D%20%3D%20)
±
![\frac{ \pi }{6} + 2 \pi n](https://tex.z-dn.net/?f=%20%5Cfrac%7B%20%5Cpi%20%7D%7B6%7D%20%2B%202%20%20%5Cpi%20n)
, n ∈ Z
![x =](https://tex.z-dn.net/?f=x%20%3D%20)
±
![\frac{ \pi }{3} + \frac{ \pi }{2} + 4 \pi n](https://tex.z-dn.net/?f=%5Cfrac%7B%20%5Cpi%20%7D%7B3%7D%20%2B%20%20%20%5Cfrac%7B%20%5Cpi%20%7D%7B2%7D%20%2B%204%20%5Cpi%20n)
, n ∈ Z
1)![9x{^2}+12x=5](https://tex.z-dn.net/?f=9x%7B%5E2%7D%2B12x%3D5)
![9x{^2}+12x-5=0](https://tex.z-dn.net/?f=9x%7B%5E2%7D%2B12x-5%3D0)
![D=12{^2}-4*9*(-5)=144+180=324>0](https://tex.z-dn.net/?f=D%3D12%7B%5E2%7D-4%2A9%2A%28-5%29%3D144%2B180%3D324%3E0)
![x{_1}=\frac{-12+18}{18}=\frac{6}{18}=\frac{1}{3}](https://tex.z-dn.net/?f=x%7B_1%7D%3D%5Cfrac%7B-12%2B18%7D%7B18%7D%3D%5Cfrac%7B6%7D%7B18%7D%3D%5Cfrac%7B1%7D%7B3%7D)
![x{_2}=\frac{-12-18}{18}=-\frac{30}{18}=-\frac{5}{3}](https://tex.z-dn.net/?f=x%7B_2%7D%3D%5Cfrac%7B-12-18%7D%7B18%7D%3D-%5Cfrac%7B30%7D%7B18%7D%3D-%5Cfrac%7B5%7D%7B3%7D)
![|x{_1}-x{_2}|=|\frac{1}{3}-(-\frac{5}{3})|=|\frac{1}{3}+\frac{5}{3}|=|\frac{6}{3}|=|2|=2](https://tex.z-dn.net/?f=%7Cx%7B_1%7D-x%7B_2%7D%7C%3D%7C%5Cfrac%7B1%7D%7B3%7D-%28-%5Cfrac%7B5%7D%7B3%7D%29%7C%3D%7C%5Cfrac%7B1%7D%7B3%7D%2B%5Cfrac%7B5%7D%7B3%7D%7C%3D%7C%5Cfrac%7B6%7D%7B3%7D%7C%3D%7C2%7C%3D2)
Ответ: 2.
2) 1) ![-24x{^2}+38x-15=0](https://tex.z-dn.net/?f=-24x%7B%5E2%7D%2B38x-15%3D0)
![24x{^2}-38x+15=0](https://tex.z-dn.net/?f=24x%7B%5E2%7D-38x%2B15%3D0)
![\frac{D}{4}=(-19){^2}-24*15=361-360=1>0](https://tex.z-dn.net/?f=%5Cfrac%7BD%7D%7B4%7D%3D%28-19%29%7B%5E2%7D-24%2A15%3D361-360%3D1%3E0)
![x{_1}=\frac{19+1}{24}=\frac{20}{24}=\frac{5}{6}](https://tex.z-dn.net/?f=x%7B_1%7D%3D%5Cfrac%7B19%2B1%7D%7B24%7D%3D%5Cfrac%7B20%7D%7B24%7D%3D%5Cfrac%7B5%7D%7B6%7D)
![x{_2}=\frac{19-1}{24}=\frac{18}{24}=\frac{3}{4}](https://tex.z-dn.net/?f=x%7B_2%7D%3D%5Cfrac%7B19-1%7D%7B24%7D%3D%5Cfrac%7B18%7D%7B24%7D%3D%5Cfrac%7B3%7D%7B4%7D)
2)
НОЗ=12
![\frac{10}{12}>\frac{9}{12}](https://tex.z-dn.net/?f=%5Cfrac%7B10%7D%7B12%7D%3E%5Cfrac%7B9%7D%7B12%7D)
Ответ: наименьший корень
.