Mister Exam
Other calculators
-
How to use it?
- Equation:
-
Equation -4x+9=-19
- Equation 13+(y+1)^2=10+(-2-y)^2
- Equation x(x-2)+(x-2)(x+4)=2
- Equation x^4-3x^2+2x-5/(|x|-2)=4x-2y+5
- Express {x} in terms of y where:
- 18*x-4*y=5
- 11*x+2*y=-2
- -19*x-15*y=12
- 2*x-16*y=19
-
Identical expressions
- sin(z)+i= zero
- sinus of (z) plus i equally 0
- sinus of (z) plus i equally zero
- sinz+i=0
- sin(z)+i=O
-
Similar expressions
- sin(z)-i=0
sin(z)+i=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation
$$\sin{\left(z \right)} + i = 0$$
- this is the simplest trigonometric equation
We get:
$$\sin{\left(z \right)} = - i$$
This equation is transformed to
$$z = 2 \pi n + \operatorname{asin}{\left(- i \right)}$$
$$z = 2 \pi n + \pi - \operatorname{asin}{\left(- i \right)}$$
Or
$$z = 2 \pi n - i \log{\left(1 + \sqrt{2} \right)}$$
$$z = 2 \pi n + \pi + i \log{\left(1 + \sqrt{2} \right)}$$
, where n - is a integer
$$\sin{\left(z \right)} + i = 0$$
- this is the simplest trigonometric equation
Move i to right part of the equation
with the change of sign in i
We get:
$$\sin{\left(z \right)} = - i$$
This equation is transformed to
$$z = 2 \pi n + \operatorname{asin}{\left(- i \right)}$$
$$z = 2 \pi n + \pi - \operatorname{asin}{\left(- i \right)}$$
Or
$$z = 2 \pi n - i \log{\left(1 + \sqrt{2} \right)}$$
$$z = 2 \pi n + \pi + i \log{\left(1 + \sqrt{2} \right)}$$
, where n - is a integer
The graph
Sum and product of roots
[src]
sum
/ ___\ / ___\ - I*log\1 + \/ 2 / + pi + I*log\1 + \/ 2 /
$$- i \log{\left(1 + \sqrt{2} \right)} + \left(\pi + i \log{\left(1 + \sqrt{2} \right)}\right)$$
=
pi
$$\pi$$
product
/ ___\ / / ___\\ -I*log\1 + \/ 2 /*\pi + I*log\1 + \/ 2 //
$$- i \log{\left(1 + \sqrt{2} \right)} \left(\pi + i \log{\left(1 + \sqrt{2} \right)}\right)$$
=
/ / ___\\ / ___\ \-pi*I + log\1 + \/ 2 //*log\1 + \/ 2 /
$$\left(\log{\left(1 + \sqrt{2} \right)} - i \pi\right) \log{\left(1 + \sqrt{2} \right)}$$
(-pi*i + log(1 + sqrt(2)))*log(1 + sqrt(2))
Rapid solution
[src]
/ ___\ z1 = -I*log\1 + \/ 2 /
$$z_{1} = - i \log{\left(1 + \sqrt{2} \right)}$$
/ ___\ z2 = pi + I*log\1 + \/ 2 /
$$z_{2} = \pi + i \log{\left(1 + \sqrt{2} \right)}$$
z2 = pi + i*log(1 + sqrt(2))
Numerical answer
[src]
z1 = 53.4070751110265 + 0.881373587019543*i
z2 = -78.5398163397448 + 0.881373587019543*i
z3 = 87.9645943005142 - 0.881373587019543*i
z4 = 34.5575191894877 + 0.881373587019543*i
z5 = 40.8407044966673 + 0.881373587019543*i
z6 = -0.881373587019543*i
z7 = -65.9734457253857 + 0.881373587019543*i
z8 = -43.9822971502571 - 0.881373587019543*i
z9 = -69.1150383789755 - 0.881373587019543*i
z10 = -40.8407044966673 + 0.881373587019543*i
z11 = -37.6991118430775 - 0.881373587019543*i
z12 = 81.6814089933346 - 0.881373587019543*i
z13 = -50.2654824574367 - 0.881373587019543*i
z14 = -3.14159265358979 + 0.881373587019543*i
z15 = -87.9645943005142 - 0.881373587019543*i
z16 = -56.5486677646163 - 0.881373587019543*i
z17 = 94.2477796076938 - 0.881373587019543*i
z18 = 78.5398163397448 + 0.881373587019543*i
z19 = -18.8495559215388 - 0.881373587019543*i
z20 = 37.6991118430775 - 0.881373587019543*i
z21 = 9.42477796076938 + 0.881373587019543*i
z22 = 21.9911485751286 + 0.881373587019543*i
z23 = -75.398223686155 - 0.881373587019543*i
z24 = -21.9911485751286 + 0.881373587019543*i
z25 = 6.28318530717959 - 0.881373587019543*i
z26 = -6.28318530717959 - 0.881373587019543*i
z27 = -84.8230016469244 + 0.881373587019543*i
z28 = 56.5486677646163 - 0.881373587019543*i
z29 = 12.5663706143592 - 0.881373587019543*i
z30 = -97.3893722612836 + 0.881373587019543*i
z31 = 97.3893722612836 + 0.881373587019543*i
z32 = 15.707963267949 + 0.881373587019543*i
z33 = 50.2654824574367 - 0.881373587019543*i
z34 = -62.8318530717959 - 0.881373587019543*i
z35 = 84.8230016469244 + 0.881373587019543*i
z36 = -15.707963267949 + 0.881373587019543*i
z37 = 28.2743338823081 + 0.881373587019543*i
z38 = -34.5575191894877 + 0.881373587019543*i
z39 = 3.14159265358979 + 0.881373587019543*i
z40 = -94.2477796076938 - 0.881373587019543*i
z41 = 69.1150383789755 - 0.881373587019543*i
z42 = -53.4070751110265 + 0.881373587019543*i
z43 = -28.2743338823081 + 0.881373587019543*i
z44 = -72.2566310325652 + 0.881373587019543*i
z45 = 43.9822971502571 - 0.881373587019543*i
z46 = -100.530964914873 - 0.881373587019543*i
z47 = -59.6902604182061 + 0.881373587019543*i
z48 = -31.4159265358979 - 0.881373587019543*i
z49 = -25.1327412287183 - 0.881373587019543*i
z50 = 75.398223686155 - 0.881373587019543*i
z51 = -81.6814089933346 - 0.881373587019543*i
z52 = 25.1327412287183 - 0.881373587019543*i
z53 = 65.9734457253857 + 0.881373587019543*i
z54 = 72.2566310325652 + 0.881373587019543*i
z55 = -9.42477796076938 + 0.881373587019543*i
z56 = -12.5663706143592 - 0.881373587019543*i
z57 = 59.6902604182061 + 0.881373587019543*i
z58 = 62.8318530717959 - 0.881373587019543*i
z59 = 18.8495559215388 - 0.881373587019543*i
z60 = 47.1238898038469 + 0.881373587019543*i
z61 = -47.1238898038469 + 0.881373587019543*i
z62 = -91.106186954104 + 0.881373587019543*i
z63 = 100.530964914873 - 0.881373587019543*i
z64 = 91.106186954104 + 0.881373587019543*i
z65 = 31.4159265358979 - 0.881373587019543*i
z65 = 31.4159265358979 - 0.881373587019543*i