Благоприятных условий - 4
Всего возможных ситуаций - 66
Вероятность = Благоприятные ситуации / Все ситуации = 4/66
Используем формулу площади треугольника S=1/2absinC, где a и b стороны треугольника, а С угол между ними.
SBOD=1/2OB*ODsinO
SAOC=1/2OA*OCsinO
Разделим первое уравнение на второе
SBOD/SAOC = (OB*OD)/(OA*OC)
Учтем, что OB=OC и OD=3OA
SBOD/SAOC = (3OA*OC)/(OA*OC)=3
SBOD = 3SAOC=3*16=48квсм
![1) \: \: 4 {sin}^{2} x - sin2x = 3 \\ 4 {sin}^{2} x - sin2x - 3 = 0 \\ 4 {sin}^{2} x - 2sinxcosx - 3( {sin}^{2} x + {cos}^{2} x) = 0 \\ 4 {sin}^{2} x - 2sinxcosx - 3 {sin}^{2} x - 3 {cos}^{2} x = 0 \\ {sin}^{2} x - 2sinxcosx - 3 {cos}^{2} x = 0 \\ \frac{ {sin}^{2} x}{ {cos}^{2} x} - \frac{2sinxcosx}{ {cos}^{2} x} - \frac{3 {cos}^{2} x}{ {cos}^{2} x} = 0 \\ {tg}^{2} x - 2tgx - 3 = 0 \\ tgx = t \\ {t}^{2} - 2t - 3 = 0 \\ d = {b}^{2} - 4ac = 4 - 4 \times ( - 3) = 16 \\ t1 = \frac{2 + 4}{2} = 3 \\ t2 = \frac{2 - 4}{2} = - 1 \\ 1)tgx = 3 \\ x = arctg3 + \pi n \\ 2) tgx = - 1 \\ x = - \frac{\pi}{4} + \pi n](https://tex.z-dn.net/?f=1%29+%5C%3A++%5C%3A+4+%7Bsin%7D%5E%7B2%7D+x+-+sin2x+%3D+3+%5C%5C+4+%7Bsin%7D%5E%7B2%7D+x+-+sin2x+-+3+%3D+0+%5C%5C+4+%7Bsin%7D%5E%7B2%7D+x+-+2sinxcosx+-+3%28+%7Bsin%7D%5E%7B2%7D+x+%2B++%7Bcos%7D%5E%7B2%7D+x%29+%3D+0+%5C%5C+4+%7Bsin%7D%5E%7B2%7D+x+-+2sinxcosx+-+3+%7Bsin%7D%5E%7B2%7D+x+-+3+%7Bcos%7D%5E%7B2%7D+x+%3D+0+%5C%5C++%7Bsin%7D%5E%7B2%7D+x+-+2sinxcosx+-+3+%7Bcos%7D%5E%7B2%7D+x+%3D+0+%5C%5C++%5Cfrac%7B+%7Bsin%7D%5E%7B2%7D+x%7D%7B+%7Bcos%7D%5E%7B2%7D+x%7D++-++%5Cfrac%7B2sinxcosx%7D%7B+%7Bcos%7D%5E%7B2%7D+x%7D++-++%5Cfrac%7B3+%7Bcos%7D%5E%7B2%7D+x%7D%7B+%7Bcos%7D%5E%7B2%7D+x%7D++%3D+0+%5C%5C++%7Btg%7D%5E%7B2%7D+x+-+2tgx+-+3+%3D+0+%5C%5C+tgx+%3D+t+%5C%5C++%7Bt%7D%5E%7B2%7D++-+2t+-+3+%3D+0+%5C%5C+d+%3D++%7Bb%7D%5E%7B2%7D++-+4ac+%3D+4+-+4+%5Ctimes+%28+-+3%29+%3D+16+%5C%5C+t1+%3D++%5Cfrac%7B2+%2B+4%7D%7B2%7D++%3D+3+%5C%5C+t2+%3D++%5Cfrac%7B2+-+4%7D%7B2%7D++%3D++-+1+%5C%5C+1%29tgx+%3D+3+%5C%5C+x+%3D+arctg3+%2B+%5Cpi+n+%5C%5C+2%29+tgx+%3D++-+1+%5C%5C++x+%3D++-++%5Cfrac%7B%5Cpi%7D%7B4%7D++%2B+%5Cpi+n)
Ответ; arctg3 + pi*n; -pi/4 + pi*n, n € Z.
2)
![sin2x + 8 {sin}^{2} x = 5 \\ sin2x + 8 {sin}^{2} x - 5 = 0 \\ 2sinxcosx + 8 {sin}^{2} x - 5( {sin}^{2} x + {cos}^{2} x) = 0 \\2sinxcosx + 8 {sin}^{2} x - 5 {sin}^{2} x - 5 {cos}^{2} x = 0 \\ 3 {sin}^{2} x + 2sinxcosx - 5 {cos}^{2} x = 0 \\ \frac{3 {sin}^{2} x}{ {cos}^{2} x} + \frac{2sinxcosx}{ {cos}^{2}x } - \frac{5 {cos}^{2} x}{ {cos}^{2}x } = 0 \\ 3 {tg}^{2} x + 2tgx - 5 = 0 \\ tgx = t \\ 3 {t}^{2} + 2t - 5 = 0 \\ d = {b}^{2} - 4ac = 4 - 4 \times 3 \times ( - 5) = 64 \\ t1 = \frac{ - 2 + 8}{2 \times 3} = \frac{6}{6} = 1 \\ t2 = \frac{ - 2 - 8}{2 \times 3} = \frac{ - 10}{6} = - \frac{5}{3} \\ 1)tgx = 1 \\ x = \frac{\pi}{4} + \pi n \\ 2)tgx = - \frac{5}{3} \\ x = - arctg \frac{5}{3} + \pi n](https://tex.z-dn.net/?f=sin2x+%2B+8+%7Bsin%7D%5E%7B2%7D+x+%3D+5+%5C%5C+sin2x+%2B+8+%7Bsin%7D%5E%7B2%7D+x+-+5+%3D+0+%5C%5C+2sinxcosx+%2B+8+%7Bsin%7D%5E%7B2%7D+x+-+5%28+%7Bsin%7D%5E%7B2%7D+x+%2B++%7Bcos%7D%5E%7B2%7D+x%29+%3D+0+%5C%5C2sinxcosx+%2B+8+%7Bsin%7D%5E%7B2%7D+x+-+5+%7Bsin%7D%5E%7B2%7D+x+-+5+%7Bcos%7D%5E%7B2%7D+x+%3D+0+%5C%5C+3+%7Bsin%7D%5E%7B2%7D+x+%2B+2sinxcosx+-+5+%7Bcos%7D%5E%7B2%7D+x+%3D+0+%5C%5C++%5Cfrac%7B3+%7Bsin%7D%5E%7B2%7D+x%7D%7B+%7Bcos%7D%5E%7B2%7D+x%7D++%2B++%5Cfrac%7B2sinxcosx%7D%7B+%7Bcos%7D%5E%7B2%7Dx+%7D++-++%5Cfrac%7B5+%7Bcos%7D%5E%7B2%7D+x%7D%7B+%7Bcos%7D%5E%7B2%7Dx+%7D++%3D+0+%5C%5C+3+%7Btg%7D%5E%7B2%7D+x+%2B+2tgx+-+5+%3D+0+%5C%5C+tgx+%3D+t+%5C%5C+3+%7Bt%7D%5E%7B2%7D++%2B+2t+-+5+%3D+0+%5C%5C+d+%3D++%7Bb%7D%5E%7B2%7D++-+4ac+%3D+4+-+4+%5Ctimes+3+%5Ctimes+%28+-+5%29+%3D+64+%5C%5C+t1+%3D++%5Cfrac%7B+-+2+%2B+8%7D%7B2+%5Ctimes+3%7D++%3D++%5Cfrac%7B6%7D%7B6%7D++%3D+1+%5C%5C+t2+%3D++%5Cfrac%7B+-+2+-+8%7D%7B2+%5Ctimes+3%7D++%3D++%5Cfrac%7B+-+10%7D%7B6%7D++%3D++-++%5Cfrac%7B5%7D%7B3%7D++%5C%5C+1%29tgx+%3D+1+%5C%5C+x+%3D++%5Cfrac%7B%5Cpi%7D%7B4%7D++%2B+%5Cpi+n+%5C%5C+2%29tgx+%3D++-++%5Cfrac%7B5%7D%7B3%7D++%5C%5C+x+%3D++-+arctg+%5Cfrac%7B5%7D%7B3%7D++%2B+%5Cpi+n)
Ответ: pi/4 + pi*n; -arctg5/3 + pi*n, n € Z.
3)
![10 {cos}^{2} x - 2sin2x = 3 \\ 10 {cos}^{2} x - 4sin2x - 3 = 0 \\ 10 {cos}^{2} x - 4sinxcosx - 3( {sin}^{2} x + {cos}^{2} x ) = 0 \\ 10 {cos}^{2} x - 4sinxcosx - 3 {sin}^{2} x - 3 {cos}^{2} x = 0 \\ 7 {cos}^{2} x - 4sinxcosx - 3 {sin}^{2} x = 0 \\ \frac{7 {cos}^{2} x}{ {cos}^{2} x} - \frac{4sinxcosx}{ {cos}^{2}x } - \frac{3 {sin}^{2} x}{ {cos}^{2} x} = 0 \\ 7 - 4tgx - 3 {tg}^{2} x = 0 \\ 3 {tg}^{2} x + 4tgx - 7 = 0 \\ tgx = t \\ 3 {t}^{2} + 4t - 7 = 0 \\ d = {b}^{2} - 4ac = 16 - 4 \times 3 \times ( - 7 ) = 100 \\ t1 = \frac{ - 4 + 10}{2 \times 3} = \frac{6}{6} = 1 \\ t2 = \frac{ - 4 - 10}{2 \times 3} = \frac{ - 14}{6} = - \frac{7}{3} \\ 1)tgx = 1 \\ x = \frac{\pi}{4} + \pi n \\ 2)tgx = - \frac{7}{3} \\ x = - arctg \frac{7}{3} + \pi n](https://tex.z-dn.net/?f=10+%7Bcos%7D%5E%7B2%7D+x+-+2sin2x+%3D+3+%5C%5C+10+%7Bcos%7D%5E%7B2%7D+x+-+4sin2x+-+3+%3D+0+%5C%5C+10+%7Bcos%7D%5E%7B2%7D+x+-+4sinxcosx+-+3%28+%7Bsin%7D%5E%7B2%7D+x+%2B++%7Bcos%7D%5E%7B2%7D+x+%29+%3D+0+%5C%5C+10+%7Bcos%7D%5E%7B2%7D+x+-+4sinxcosx+-+3+%7Bsin%7D%5E%7B2%7D+x+-+3+%7Bcos%7D%5E%7B2%7D+x+%3D+0+%5C%5C+7+%7Bcos%7D%5E%7B2%7D+x+-+4sinxcosx++-+3+%7Bsin%7D%5E%7B2%7D+x+%3D+0+%5C%5C++%5Cfrac%7B7+%7Bcos%7D%5E%7B2%7D+x%7D%7B+%7Bcos%7D%5E%7B2%7D+x%7D++-++%5Cfrac%7B4sinxcosx%7D%7B+%7Bcos%7D%5E%7B2%7Dx+%7D++-++%5Cfrac%7B3+%7Bsin%7D%5E%7B2%7D+x%7D%7B+%7Bcos%7D%5E%7B2%7D+x%7D++%3D+0+%5C%5C+7+-+4tgx+-+3+%7Btg%7D%5E%7B2%7D+x+%3D+0+%5C%5C+3+%7Btg%7D%5E%7B2%7D+x+%2B+4tgx+-+7+%3D+0+%5C%5C+tgx+%3D+t+%5C%5C+3+%7Bt%7D%5E%7B2%7D++%2B+4t+-+7+%3D+0+%5C%5C+d+%3D++%7Bb%7D%5E%7B2%7D++-+4ac+%3D+16+-+4+%5Ctimes+3+%5Ctimes+%28+-+7+%29+%3D+100+%5C%5C+t1+%3D++%5Cfrac%7B+-+4+%2B+10%7D%7B2+%5Ctimes+3%7D++%3D++%5Cfrac%7B6%7D%7B6%7D++%3D+1+%5C%5C+t2+%3D++%5Cfrac%7B+-+4+-+10%7D%7B2+%5Ctimes+3%7D++%3D++%5Cfrac%7B+-+14%7D%7B6%7D++%3D+++-++%5Cfrac%7B7%7D%7B3%7D++%5C%5C+1%29tgx+%3D+1+%5C%5C+x+%3D++%5Cfrac%7B%5Cpi%7D%7B4%7D++%2B+%5Cpi+n+%5C%5C+2%29tgx+%3D++-++%5Cfrac%7B7%7D%7B3%7D++%5C%5C+x+%3D++-+arctg+%5Cfrac%7B7%7D%7B3%7D++%2B+%5Cpi+n)
Ответ: pi/4 + pi*n; -arctg7/3 + pi*n, n € Z.
4х(х+3)=4-3х 4х²+12х=4-3х. 4х²+12х-4+3х=0. 4х²+15х-4=0. Д: 225+64= 289 = 17² х1 = (-15+7):8= -1; х2= (-15-7):8 = -2.75
A) 3/8
b)8/15y
в)-7y/9
г)x/2
2)x≠0 x≠1