sin(z)=3i/4 (уравнение)

Найду корень уравнения: sin(z)=3i/4

Вы ввели

         3*I
sin(z) = ---
          4 

$$\sin{\left(z \right)} = \frac{3 i}{4}$$

Подробное решение

Дано уравнение
$$\sin{\left(z \right)} = \frac{3 i}{4}$$
- это простейшее тригонометрическое ур-ние
Это ур-ние преобразуется в
$$z = 2 \pi n + \operatorname{asin}{\left(\frac{3 i}{4} \right)}$$
$$z = 2 \pi n + \pi - \operatorname{asin}{\left(\frac{3 i}{4} \right)}$$
Или
$$z = 2 \pi n + i \operatorname{asinh}{\left(\frac{3}{4} \right)}$$
$$z = 2 \pi n + \pi - i \operatorname{asinh}{\left(\frac{3}{4} \right)}$$
, где n - любое целое число

Быстрый ответ

z1 = I*asinh(3/4)

$$z_{1} = i \operatorname{asinh}{\left(\frac{3}{4} \right)}$$

z2 = pi - I*asinh(3/4)

$$z_{2} = \pi - i \operatorname{asinh}{\left(\frac{3}{4} \right)}$$

Сумма и произведение корней

сумма

I*asinh(3/4) + pi - I*asinh(3/4)

$$\left(\pi - i \operatorname{asinh}{\left(\frac{3}{4} \right)}\right) + i \operatorname{asinh}{\left(\frac{3}{4} \right)}$$

=

pi

$$\pi$$

произведение

I*asinh(3/4)*(pi - I*asinh(3/4))

$$i \operatorname{asinh}{\left(\frac{3}{4} \right)} \left(\pi - i \operatorname{asinh}{\left(\frac{3}{4} \right)}\right)$$

=

(pi*I + asinh(3/4))*asinh(3/4)

$$\left(\operatorname{asinh}{\left(\frac{3}{4} \right)} + i \pi\right) \operatorname{asinh}{\left(\frac{3}{4} \right)}$$

Численный ответ

z1 = -72.2566310325652 - 0.693147180559945*i

z2 = -62.8318530717959 + 0.693147180559945*i

z3 = -65.9734457253857 - 0.693147180559945*i

z4 = 47.1238898038469 - 0.693147180559945*i

z5 = 69.1150383789755 + 0.693147180559945*i

z6 = -53.4070751110265 - 0.693147180559945*i

z7 = -47.1238898038469 - 0.693147180559945*i

z8 = -75.398223686155 + 0.693147180559945*i

z9 = -78.5398163397448 - 0.693147180559945*i

z10 = -25.1327412287183 + 0.693147180559945*i

z11 = 15.707963267949 - 0.693147180559945*i

z12 = 75.398223686155 + 0.693147180559945*i

z13 = -56.5486677646163 + 0.693147180559945*i

z14 = -50.2654824574367 + 0.693147180559945*i

z15 = 50.2654824574367 + 0.693147180559945*i

z16 = -15.707963267949 - 0.693147180559945*i

z17 = 37.6991118430775 + 0.693147180559945*i

z18 = -3.14159265358979 - 0.693147180559945*i

z19 = 100.530964914873 + 0.693147180559945*i

z20 = -87.9645943005142 + 0.693147180559945*i

z21 = 72.2566310325652 - 0.693147180559945*i

z22 = -9.42477796076938 - 0.693147180559945*i

z23 = 94.2477796076938 + 0.693147180559945*i

z24 = 40.8407044966673 - 0.693147180559945*i

z25 = 91.106186954104 - 0.693147180559945*i

z26 = 62.8318530717959 + 0.693147180559945*i

z27 = 9.42477796076938 - 0.693147180559945*i

z28 = -81.6814089933346 + 0.693147180559945*i

z29 = 81.6814089933346 + 0.693147180559945*i

z30 = -12.5663706143592 + 0.693147180559945*i

z31 = 78.5398163397448 - 0.693147180559945*i

z32 = 59.6902604182061 - 0.693147180559945*i

z33 = -37.6991118430775 + 0.693147180559945*i

z34 = -21.9911485751286 - 0.693147180559945*i

z35 = -34.5575191894877 - 0.693147180559945*i

z36 = 84.8230016469244 - 0.693147180559945*i

z37 = -94.2477796076938 + 0.693147180559945*i

z38 = 18.8495559215388 + 0.693147180559945*i

z39 = -28.2743338823081 - 0.693147180559945*i

z40 = 0.693147180559945*i

z41 = 65.9734457253857 - 0.693147180559945*i

z42 = -100.530964914873 + 0.693147180559945*i

z43 = -84.8230016469244 - 0.693147180559945*i

z44 = 28.2743338823081 - 0.693147180559945*i

z45 = -43.9822971502571 + 0.693147180559945*i

z46 = -97.3893722612836 - 0.693147180559945*i

z47 = -91.106186954104 - 0.693147180559945*i

z48 = 12.5663706143592 + 0.693147180559945*i

z49 = -59.6902604182061 - 0.693147180559945*i

z50 = 21.9911485751286 - 0.693147180559945*i

z51 = 53.4070751110265 - 0.693147180559945*i

z52 = -18.8495559215388 + 0.693147180559945*i

z53 = -31.4159265358979 + 0.693147180559945*i

z54 = -69.1150383789755 + 0.693147180559945*i

z55 = 31.4159265358979 + 0.693147180559945*i

z56 = 43.9822971502571 + 0.693147180559945*i

z57 = 56.5486677646163 + 0.693147180559945*i

z58 = -6.28318530717959 + 0.693147180559945*i

z59 = 3.14159265358979 - 0.693147180559945*i

z60 = 25.1327412287183 + 0.693147180559945*i

z61 = 34.5575191894877 - 0.693147180559945*i

z62 = 97.3893722612836 - 0.693147180559945*i

z63 = 87.9645943005142 + 0.693147180559945*i

z64 = -40.8407044966673 - 0.693147180559945*i

z65 = 6.28318530717959 + 0.693147180559945*i