sin(x)/cos(x)=6 (уравнение)

Найду корень уравнения: sin(x)/cos(x)=6

Вы ввели

[LaTeX]

sin(x)    
------ = 6
cos(x)    

$$\frac{\sin{\left (x \right )}}{\cos{\left (x \right )}} = 6$$

Подробное решение

[LaTeX]

Дано уравнение
$$\frac{\sin{\left (x \right )}}{\cos{\left (x \right )}} = 6$$
преобразуем
$$\tan{\left (x \right )} - 6 = 0$$
$$\tan{\left (x \right )} - 6 = 0$$
Сделаем замену
$$w = \tan{\left (x \right )}$$
Переносим свободные слагаемые (без w)
из левой части в правую, получим:
$$w = 6$$
Получим ответ: w = 6
делаем обратную замену
$$\tan{\left (x \right )} = w$$
Дано уравнение
$$\tan{\left (x \right )} = w$$
- это простейшее тригонометрическое ур-ние
Это ур-ние преобразуется в
$$x = \pi n + \operatorname{atan}{\left (w \right )}$$
Или
$$x = \pi n + \operatorname{atan}{\left (w \right )}$$
, где n - любое целое число
подставляем w:
$$x_{1} = \pi n + \operatorname{atan}{\left (w_{1} \right )}$$
$$x_{1} = \pi n + \operatorname{atan}{\left (6 \right )}$$
$$x_{1} = \pi n + \operatorname{atan}{\left (6 \right )}$$

Быстрый ответ

[LaTeX]

            /      ____\
            |1   \/ 37 |
x1 = -2*atan|- + ------|
            \6     6   /

$$x_{1} = - 2 \operatorname{atan}{\left (\frac{1}{6} + \frac{\sqrt{37}}{6} \right )}$$

            /      ____\
            |1   \/ 37 |
x2 = -2*atan|- - ------|
            \6     6   /

$$x_{2} = - 2 \operatorname{atan}{\left (- \frac{\sqrt{37}}{6} + \frac{1}{6} \right )}$$

Численный ответ

[LaTeX]

x1 = 42.2463521460000

x2 = 145.918909715000

x3 = 4.54724030297000

x4 = 13.9720182637000

x5 = -92.8421319583000

x6 = 39.1047594925000

x7 = 57.9543154140000

x8 = 17.1136109173000

x9 = 10.8304256101000

x10 = -83.4173539975000

x11 = -130.541243801000

x12 = 92.5118346035000

x13 = 29.6799815317000

x14 = -80.2757613440000

x15 = 64.2375007212000

x16 = 83.0870566427000

x17 = 45.3879447996000

x18 = 7.68883295656000

x19 = -95.9837246119000

x20 = 70.5206860284000

x21 = 26.5383888781000

x22 = 365.830395466000

x23 = -33.1518715401000

x24 = 32.8215741853000

x25 = 61.0959080676000

x26 = -4.87753765780000

x27 = -73.9925760368000

x28 = -52.0014274616000

x29 = 67.3790933748000

x30 = 73.6622786819000

x31 = 86.2286492963000

x32 = 48.5295374532000

x33 = 35.9631668389000

x34 = 79.9454639891000

x35 = -114.833280533000

x36 = -23.7270935793000

x37 = -55.1430201152000

x38 = -30.0102788865000

x39 = -58.2846127688000

x40 = -36.2934641937000

x41 = 1.40564764938000

x42 = -61.4262054224000

x43 = -86.5589466511000

x44 = 95.6534272571000

x45 = 54.8127227604000

x46 = 117.644575832000

x47 = -8.01913031139000

x48 = 23.3967962245000

x49 = 76.8038713355000

x50 = -11.1607229650000

x51 = -20.5855009257000

x52 = -14.3023156186000

x53 = -39.4350568473000

x54 = -70.8509833832000

x55 = 51.6711301068000

x56 = 89.3702419499000

x57 = -89.7005393047000

x58 = -26.8686862329000

x59 = -17.4439082722000

x60 = -48.8598348081000

x61 = 20.2552035709000

x62 = -743.151811251000

x63 = -45.7182421545000

x64 = -77.1341686904000

x65 = 98.7950199107000

x66 = -1.73594500421000

x67 = -99.1253172655000

x68 = -42.5766495009000

x69 = -64.5677980760000

x70 = -67.7093907296000