Mister Exam
Other calculators
-
How to use it?
- Equation:
-
Equation cos(x)-1=0
- Equation 64-x^2=0
-
Equation sin(5*x)=-1
-
Equation 7^x-2=49
- Express {x} in terms of y where:
- -16*x-10*y=-7
- -17*x+16*y=4
- -10*x+18*y=-10
- 16*x-9*y=-11
- Derivative of:
- sin5(x)
-
-sin(x)
- Integral of d{x}:
-
sin5(x)
- Graphing y =:
-
-sin(x)
- Limit of the function:
-
-sin(x)
-
Identical expressions
- sin5(x)=-sin(x)
- sinus of 5(x) equally minus sinus of (x)
- sin5x=-sinx
-
Similar expressions
- sin(5*x)=-1
- sin(3*x)+sin(5*x)=5sin(4*x)
- -sin(x-pi/3)=0
- sin(5*x)*cos(2*x)+sin(5*x)*sin(2*x)=0
- sin5(x)=+sin(x)
- sin(5x/3)+sin(3x/2)=0
- sin5(x)=-sinx
sin5(x)=-sin(x) equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation
$$\sin^{5}{\left(x \right)} = - \sin{\left(x \right)}$$
transform
$$\sin^{5}{\left(x \right)} + \sin{\left(x \right)} = 0$$
$$\sin^{5}{\left(x \right)} + \sin{\left(x \right)} = 0$$
Do replacement
$$w = \sin{\left(x \right)}$$
Given the equation
$$w^{5} + w = 0$$
Obviously:
next,
transform
$$\frac{1}{w^{4}} = -1$$
Because equation degree is equal to = -4 and the free term = -1 < 0,
so the real solutions of the equation d'not exist
All other 4 root(s) is the complex numbers.
do replacement:
$$z = w$$
then the equation will be the:
$$\frac{1}{z^{4}} = -1$$
Any complex number can presented so:
$$z = r e^{i p}$$
substitute to the equation
$$\frac{e^{- 4 i p}}{r^{4}} = -1$$
where
$$r = 1$$
- the magnitude of the complex number
Substitute r:
$$e^{- 4 i p} = -1$$
Using Euler’s formula, we find roots for p
$$- i \sin{\left(4 p \right)} + \cos{\left(4 p \right)} = -1$$
so
$$\cos{\left(4 p \right)} = -1$$
and
$$- \sin{\left(4 p \right)} = 0$$
then
$$p = - \frac{\pi N}{2} - \frac{\pi}{4}$$
where N=0,1,2,3,...
Looping through the values of N and substituting p into the formula for z
Consequently, the solution will be for z:
$$z_{1} = - \frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2}$$
$$z_{2} = - \frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2}$$
$$z_{3} = \frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2}$$
$$z_{4} = \frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2}$$
do backward replacement
$$z = w$$
$$w = z$$
The final answer:
$$w_{1} = - \frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2}$$
$$w_{2} = - \frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2}$$
$$w_{3} = \frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2}$$
$$w_{4} = \frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2}$$
do backward replacement
$$\sin{\left(x \right)} = w$$
Given the equation
$$\sin{\left(x \right)} = w$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
Or
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
, where n - is a integer
substitute w:
$$\sin^{5}{\left(x \right)} = - \sin{\left(x \right)}$$
transform
$$\sin^{5}{\left(x \right)} + \sin{\left(x \right)} = 0$$
$$\sin^{5}{\left(x \right)} + \sin{\left(x \right)} = 0$$
Do replacement
$$w = \sin{\left(x \right)}$$
Given the equation
$$w^{5} + w = 0$$
Obviously:
w0 = 0
next,
transform
$$\frac{1}{w^{4}} = -1$$
Because equation degree is equal to = -4 and the free term = -1 < 0,
so the real solutions of the equation d'not exist
All other 4 root(s) is the complex numbers.
do replacement:
$$z = w$$
then the equation will be the:
$$\frac{1}{z^{4}} = -1$$
Any complex number can presented so:
$$z = r e^{i p}$$
substitute to the equation
$$\frac{e^{- 4 i p}}{r^{4}} = -1$$
where
$$r = 1$$
- the magnitude of the complex number
Substitute r:
$$e^{- 4 i p} = -1$$
Using Euler’s formula, we find roots for p
$$- i \sin{\left(4 p \right)} + \cos{\left(4 p \right)} = -1$$
so
$$\cos{\left(4 p \right)} = -1$$
and
$$- \sin{\left(4 p \right)} = 0$$
then
$$p = - \frac{\pi N}{2} - \frac{\pi}{4}$$
where N=0,1,2,3,...
Looping through the values of N and substituting p into the formula for z
Consequently, the solution will be for z:
$$z_{1} = - \frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2}$$
$$z_{2} = - \frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2}$$
$$z_{3} = \frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2}$$
$$z_{4} = \frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2}$$
do backward replacement
$$z = w$$
$$w = z$$
The final answer:
w0 = 0
$$w_{1} = - \frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2}$$
$$w_{2} = - \frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2}$$
$$w_{3} = \frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2}$$
$$w_{4} = \frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2}$$
do backward replacement
$$\sin{\left(x \right)} = w$$
Given the equation
$$\sin{\left(x \right)} = w$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
Or
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
, where n - is a integer
substitute w:
Sum and product of roots
[src]
sum
/ / ___ ___\\ / / ___ ___\\ / / ___ ___\\ / / ___ ___\\ / / ___ ___\\ / / ___ ___\\ / / ___ ___\\ / / ___ ___\\ / / ___ ___\\ / / ___ ___\\ / / ___ ___\\ / / ___ ___\\ / / ___ ___\\ / / ___ ___\\ / / ___ ___\\ / / ___ ___\\
| |\/ 2 I*\/ 2 || | |\/ 2 I*\/ 2 || | |\/ 2 I*\/ 2 || | |\/ 2 I*\/ 2 || | |\/ 2 I*\/ 2 || | |\/ 2 I*\/ 2 || | |\/ 2 I*\/ 2 || | |\/ 2 I*\/ 2 || | |\/ 2 I*\/ 2 || | |\/ 2 I*\/ 2 || | |\/ 2 I*\/ 2 || | |\/ 2 I*\/ 2 || | |\/ 2 I*\/ 2 || | |\/ 2 I*\/ 2 || | |\/ 2 I*\/ 2 || | |\/ 2 I*\/ 2 ||
pi + pi - re|asin|----- - -------|| - I*im|asin|----- - -------|| + pi + I*im|asin|----- - -------|| + re|asin|----- - -------|| + pi - re|asin|----- + -------|| - I*im|asin|----- + -------|| + pi + I*im|asin|----- + -------|| + re|asin|----- + -------|| + - re|asin|----- - -------|| - I*im|asin|----- - -------|| + I*im|asin|----- - -------|| + re|asin|----- - -------|| + - re|asin|----- + -------|| - I*im|asin|----- + -------|| + I*im|asin|----- + -------|| + re|asin|----- + -------||
\ \ 2 2 // \ \ 2 2 // \ \ 2 2 // \ \ 2 2 // \ \ 2 2 // \ \ 2 2 // \ \ 2 2 // \ \ 2 2 // \ \ 2 2 // \ \ 2 2 // \ \ 2 2 // \ \ 2 2 // \ \ 2 2 // \ \ 2 2 // \ \ 2 2 // \ \ 2 2 //
$$\left(\left(- \operatorname{re}{\left(\operatorname{asin}{\left(\frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2} \right)}\right)} - i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2} \right)}\right)}\right) + \left(\left(\operatorname{re}{\left(\operatorname{asin}{\left(\frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2} \right)}\right)}\right) + \left(\left(\left(\left(- \operatorname{re}{\left(\operatorname{asin}{\left(\frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2} \right)}\right)} + \pi - i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2} \right)}\right)}\right) + \left(\left(\operatorname{re}{\left(\operatorname{asin}{\left(\frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2} \right)}\right)} + \pi + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2} \right)}\right)}\right) + \left(\pi + \left(- \operatorname{re}{\left(\operatorname{asin}{\left(\frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2} \right)}\right)} + \pi - i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2} \right)}\right)}\right)\right)\right)\right) + \left(\operatorname{re}{\left(\operatorname{asin}{\left(\frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2} \right)}\right)} + \pi + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2} \right)}\right)}\right)\right) + \left(- \operatorname{re}{\left(\operatorname{asin}{\left(\frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2} \right)}\right)} - i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2} \right)}\right)}\right)\right)\right)\right) + \left(\operatorname{re}{\left(\operatorname{asin}{\left(\frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2} \right)}\right)}\right)$$
=
5*pi
$$5 \pi$$
product
/ / / ___ ___\\ / / ___ ___\\\ / / / ___ ___\\ / / ___ ___\\\ / / / ___ ___\\ / / ___ ___\\\ / / / ___ ___\\ / / ___ ___\\\ / / / ___ ___\\ / / ___ ___\\\ / / / ___ ___\\ / / ___ ___\\\ / / / ___ ___\\ / / ___ ___\\\ / / / ___ ___\\ / / ___ ___\\\
| | |\/ 2 I*\/ 2 || | |\/ 2 I*\/ 2 ||| | | |\/ 2 I*\/ 2 || | |\/ 2 I*\/ 2 ||| | | |\/ 2 I*\/ 2 || | |\/ 2 I*\/ 2 ||| | | |\/ 2 I*\/ 2 || | |\/ 2 I*\/ 2 ||| | | |\/ 2 I*\/ 2 || | |\/ 2 I*\/ 2 ||| | | |\/ 2 I*\/ 2 || | |\/ 2 I*\/ 2 ||| | | |\/ 2 I*\/ 2 || | |\/ 2 I*\/ 2 ||| | | |\/ 2 I*\/ 2 || | |\/ 2 I*\/ 2 |||
0*pi*|pi - re|asin|----- - -------|| - I*im|asin|----- - -------|||*|pi + I*im|asin|----- - -------|| + re|asin|----- - -------|||*|pi - re|asin|----- + -------|| - I*im|asin|----- + -------|||*|pi + I*im|asin|----- + -------|| + re|asin|----- + -------|||*|- re|asin|----- - -------|| - I*im|asin|----- - -------|||*|I*im|asin|----- - -------|| + re|asin|----- - -------|||*|- re|asin|----- + -------|| - I*im|asin|----- + -------|||*|I*im|asin|----- + -------|| + re|asin|----- + -------|||
\ \ \ 2 2 // \ \ 2 2 /// \ \ \ 2 2 // \ \ 2 2 /// \ \ \ 2 2 // \ \ 2 2 /// \ \ \ 2 2 // \ \ 2 2 /// \ \ \ 2 2 // \ \ 2 2 /// \ \ \ 2 2 // \ \ 2 2 /// \ \ \ 2 2 // \ \ 2 2 /// \ \ \ 2 2 // \ \ 2 2 ///
$$0 \pi \left(- \operatorname{re}{\left(\operatorname{asin}{\left(\frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2} \right)}\right)} + \pi - i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{asin}{\left(\frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2} \right)}\right)} + \pi + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2} \right)}\right)}\right) \left(- \operatorname{re}{\left(\operatorname{asin}{\left(\frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2} \right)}\right)} + \pi - i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{asin}{\left(\frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2} \right)}\right)} + \pi + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2} \right)}\right)}\right) \left(- \operatorname{re}{\left(\operatorname{asin}{\left(\frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2} \right)}\right)} - i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{asin}{\left(\frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2} \right)}\right)}\right) \left(- \operatorname{re}{\left(\operatorname{asin}{\left(\frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2} \right)}\right)} - i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{asin}{\left(\frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2} \right)}\right)}\right)$$
=
0
$$0$$
0
Rapid solution
[src]
x1 = 0
$$x_{1} = 0$$
x2 = pi
$$x_{2} = \pi$$
/ / ___ ___\\ / / ___ ___\\
| |\/ 2 I*\/ 2 || | |\/ 2 I*\/ 2 ||
x3 = pi - re|asin|----- - -------|| - I*im|asin|----- - -------||
\ \ 2 2 // \ \ 2 2 //
$$x_{3} = - \operatorname{re}{\left(\operatorname{asin}{\left(\frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2} \right)}\right)} + \pi - i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2} \right)}\right)}$$
/ / ___ ___\\ / / ___ ___\\
| |\/ 2 I*\/ 2 || | |\/ 2 I*\/ 2 ||
x4 = pi + I*im|asin|----- - -------|| + re|asin|----- - -------||
\ \ 2 2 // \ \ 2 2 //
$$x_{4} = \operatorname{re}{\left(\operatorname{asin}{\left(\frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2} \right)}\right)} + \pi + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2} \right)}\right)}$$
/ / ___ ___\\ / / ___ ___\\
| |\/ 2 I*\/ 2 || | |\/ 2 I*\/ 2 ||
x5 = pi - re|asin|----- + -------|| - I*im|asin|----- + -------||
\ \ 2 2 // \ \ 2 2 //
$$x_{5} = - \operatorname{re}{\left(\operatorname{asin}{\left(\frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2} \right)}\right)} + \pi - i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2} \right)}\right)}$$
/ / ___ ___\\ / / ___ ___\\
| |\/ 2 I*\/ 2 || | |\/ 2 I*\/ 2 ||
x6 = pi + I*im|asin|----- + -------|| + re|asin|----- + -------||
\ \ 2 2 // \ \ 2 2 //
$$x_{6} = \operatorname{re}{\left(\operatorname{asin}{\left(\frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2} \right)}\right)} + \pi + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2} \right)}\right)}$$
/ / ___ ___\\ / / ___ ___\\
| |\/ 2 I*\/ 2 || | |\/ 2 I*\/ 2 ||
x7 = - re|asin|----- - -------|| - I*im|asin|----- - -------||
\ \ 2 2 // \ \ 2 2 //
$$x_{7} = - \operatorname{re}{\left(\operatorname{asin}{\left(\frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2} \right)}\right)} - i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2} \right)}\right)}$$
/ / ___ ___\\ / / ___ ___\\
| |\/ 2 I*\/ 2 || | |\/ 2 I*\/ 2 ||
x8 = I*im|asin|----- - -------|| + re|asin|----- - -------||
\ \ 2 2 // \ \ 2 2 //
$$x_{8} = \operatorname{re}{\left(\operatorname{asin}{\left(\frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2} \right)}\right)}$$
/ / ___ ___\\ / / ___ ___\\
| |\/ 2 I*\/ 2 || | |\/ 2 I*\/ 2 ||
x9 = - re|asin|----- + -------|| - I*im|asin|----- + -------||
\ \ 2 2 // \ \ 2 2 //
$$x_{9} = - \operatorname{re}{\left(\operatorname{asin}{\left(\frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2} \right)}\right)} - i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2} \right)}\right)}$$
/ / ___ ___\\ / / ___ ___\\
| |\/ 2 I*\/ 2 || | |\/ 2 I*\/ 2 ||
x10 = I*im|asin|----- + -------|| + re|asin|----- + -------||
\ \ 2 2 // \ \ 2 2 //
$$x_{10} = \operatorname{re}{\left(\operatorname{asin}{\left(\frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2} \right)}\right)}$$
x10 = re(asin(sqrt(2)/2 + sqrt(2)*i/2)) + i*im(asin(sqrt(2)/2 + sqrt(2)*i/2))
Numerical answer
[src]
x1 = -43.9822971502571
x2 = -31.4159265358979
x3 = 109.955742875643
x4 = 84.8230016469244
x5 = -91.106186954104
x6 = 91.106186954104
x7 = -97.3893722612836
x8 = 6.28318530717959
x9 = -72.2566310325652
x10 = -47.1238898038469
x11 = 94.2477796076938
x12 = 50.2654824574367
x13 = 56.5486677646163
x14 = 43.9822971502571
x15 = 47.1238898038469
x16 = -50.2654824574367
x17 = 37.6991118430775
x18 = -28.2743338823081
x19 = 65.9734457253857
x20 = 15.707963267949
x21 = 28.2743338823081
x22 = -62.8318530717959
x23 = 40.8407044966673
x24 = -40.8407044966673
x25 = -6.28318530717959
x26 = -81.6814089933346
x27 = -15.707963267949
x28 = -59.6902604182061
x29 = 72.2566310325652
x30 = 3.14159265358979
x31 = -25.1327412287183
x32 = 21.9911485751286
x33 = -75.398223686155
x34 = -56.5486677646163
x35 = -69.1150383789755
x36 = -84.8230016469244
x37 = 78.5398163397448
x38 = 9.42477796076938
x39 = -53.4070751110265
x40 = 62.8318530717959
x41 = -18.8495559215388
x42 = 25.1327412287183
x43 = 100.530964914873
x44 = -87.9645943005142
x45 = -9.42477796076938
x46 = 75.398223686155
x47 = 81.6814089933346
x48 = 87.9645943005142
x49 = 12.5663706143592
x50 = -34.5575191894877
x51 = 69.1150383789755
x52 = -3.14159265358979
x53 = 0.0
x54 = -21.9911485751286
x55 = -37.6991118430775
x56 = 31.4159265358979
x57 = -78.5398163397448
x58 = -12.5663706143592
x59 = -94.2477796076938
x60 = 97.3893722612836
x61 = -125.663706143592
x62 = -100.530964914873
x63 = 59.6902604182061
x64 = 53.4070751110265
x65 = 34.5575191894877
x66 = -65.9734457253857
x67 = 18.8495559215388
x67 = 18.8495559215388