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![3 {(sin3x)}^{2} + sin6x - {(cos3x)}^{2} = 0 \\ \\ 3 {(sin3x)}^{2} + 2sin3xcos3x - {(cos3x)}^{2} = 0 \\](https://tex.z-dn.net/?f=3+%7B%28sin3x%29%7D%5E%7B2%7D+%2B+sin6x+-+%7B%28cos3x%29%7D%5E%7B2%7D+%3D+0+%5C%5C+%5C%5C+3+%7B%28sin3x%29%7D%5E%7B2%7D+%2B+2sin3xcos3x+-+%7B%28cos3x%29%7D%5E%7B2%7D+%3D+0+%5C%5C+)
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Разделим обе части на ( cos3x )^2
![\frac{3 {(sin3x)}^{2} }{ {(cos3x)}^{2} } + \frac{2sin3xcos3x}{ {(cos3x)}^{2} } - \frac{ {(cos3x)}^{2} }{ {(cos3x)}^{2} } = 0 \\ \\ 3 {(tg3x)}^{2} + 2tg3x - 1 = 0 \\](https://tex.z-dn.net/?f=+%5Cfrac%7B3+%7B%28sin3x%29%7D%5E%7B2%7D+%7D%7B+%7B%28cos3x%29%7D%5E%7B2%7D+%7D+%2B+%5Cfrac%7B2sin3xcos3x%7D%7B+%7B%28cos3x%29%7D%5E%7B2%7D+%7D+-+%5Cfrac%7B+%7B%28cos3x%29%7D%5E%7B2%7D+%7D%7B+%7B%28cos3x%29%7D%5E%7B2%7D+%7D+%3D+0+%5C%5C+%5C%5C+3+%7B%28tg3x%29%7D%5E%7B2%7D+%2B+2tg3x+-+1+%3D+0+%5C%5C+)
Сделаем замену:
Пусть tg3x = a , тогда
![3 {a}^{2} + 2a - 1 = 0 \\](https://tex.z-dn.net/?f=3+%7Ba%7D%5E%7B2%7D+%2B+2a+-+1+%3D+0+%5C%5C+)
D = 2^2 - 4•3•(-1) = 4 + 12 = 16
![a_{1} = - 1 \\ a_{2} = \frac{1}{3} \\](https://tex.z-dn.net/?f=a_%7B1%7D+%3D+-+1+%5C%5C+a_%7B2%7D+%3D+%5Cfrac%7B1%7D%7B3%7D+%5C%5C+)
Обратная замена:
![1) \: \: a = - 1 \\ \\ tg3x = - 1 \\ \\ 3x = - \frac{\pi}{4} + \pi \: n \\ \\ x = - \frac{\pi}{12} + \frac{\pi \: n}{3}](https://tex.z-dn.net/?f=1%29+%5C%3A+%5C%3A+a+%3D+-+1+%5C%5C+%5C%5C+tg3x+%3D+-+1+%5C%5C+%5C%5C+3x+%3D+-+%5Cfrac%7B%5Cpi%7D%7B4%7D+%2B+%5Cpi+%5C%3A+n+%5C%5C+%5C%5C+x+%3D+-+%5Cfrac%7B%5Cpi%7D%7B12%7D+%2B+%5Cfrac%7B%5Cpi+%5C%3A+n%7D%7B3%7D+)
n принадлежит Z
![2) \: a = \frac{1}{3} \\ \\ tg3x = \frac{1}{3} \\ \\ 3x = arctg \frac{1}{3} + \pi \: k \\ \\ x = \frac{1}{3} arc tg\frac{1}{3} + \frac{\pi \: k}{3}](https://tex.z-dn.net/?f=2%29+%5C%3A+a+%3D+%5Cfrac%7B1%7D%7B3%7D+%5C%5C+%5C%5C+tg3x+%3D+%5Cfrac%7B1%7D%7B3%7D+%5C%5C+%5C%5C+3x+%3D+arctg+%5Cfrac%7B1%7D%7B3%7D+%2B+%5Cpi+%5C%3A+k+%5C%5C+%5C%5C+x+%3D+%5Cfrac%7B1%7D%7B3%7D+arc+tg%5Cfrac%7B1%7D%7B3%7D+%2B+%5Cfrac%7B%5Cpi+%5C%3A+k%7D%7B3%7D+)
k принадлежит Z
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ОТВЕТ:
![x = - \frac{\pi}{12} + \frac{\pi \: n}{3}](https://tex.z-dn.net/?f=x+%3D+-+%5Cfrac%7B%5Cpi%7D%7B12%7D+%2B+%5Cfrac%7B%5Cpi+%5C%3A+n%7D%7B3%7D+)
![x = \frac{1}{3} arc tg\frac{1}{3} + \frac{\pi \: k}{3}](https://tex.z-dn.net/?f=+x+%3D+%5Cfrac%7B1%7D%7B3%7D+arc+tg%5Cfrac%7B1%7D%7B3%7D+%2B+%5Cfrac%7B%5Cpi+%5C%3A+k%7D%7B3%7D+)
n , k принадлежат Z
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4.8*-3/4-5*-6.4=0
48/10-3/4+32=0
81/20(4.05)+32=0
36.05
-sin(-45)=-(-sin(45))=sin45=√2/2.