Mister Exam
Other calculators
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How to use it?
- Equation:
-
Equation 4x+5=2x-7
-
Equation x^2/(x+6)=1/2
-
Equation 3-x=1+x
-
Equation 80-x=9*5
- Express {x} in terms of y where:
- -6*x+1*y=-5
- -8*x+3*y=-4
- 12*x+1*y=15
- 20*x-16*y=7
-
Identical expressions
- 27cos^3x– eight = zero
- 27 co sinus of e of cubed x–8 equally 0
- 27 co sinus of e of cubed x– eight equally zero
- 27cos3x–8=0
- 27cos³x–8=0
- 27cos to the power of 3x–8=0
- 27cos^3x–8=O
27cos^3x–8=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation
$$27 \cos^{3}{\left(x \right)} - 8 = 0$$
transform
$$27 \cos^{3}{\left(x \right)} - 8 = 0$$
$$27 \cos^{3}{\left(x \right)} - 8 = 0$$
Do replacement
$$w = \cos{\left(x \right)}$$
Given the equation
$$27 w^{3} - 8 = 0$$
Because equation degree is equal to = 3 - does not contain even numbers in the numerator, then
the equation has single real root.
Get the root 3-th degree of the equation sides:
We get:
$$\sqrt[3]{27} \sqrt[3]{w^{3}} = \sqrt[3]{8}$$
or
$$3 w = 2$$
Divide both parts of the equation by 3
We get the answer: w = 2/3
All other 2 root(s) is the complex numbers.
do replacement:
$$z = w$$
then the equation will be the:
$$z^{3} = \frac{8}{27}$$
Any complex number can presented so:
$$z = r e^{i p}$$
substitute to the equation
$$r^{3} e^{3 i p} = \frac{8}{27}$$
where
$$r = \frac{2}{3}$$
- the magnitude of the complex number
Substitute r:
$$e^{3 i p} = 1$$
Using Euler’s formula, we find roots for p
$$i \sin{\left(3 p \right)} + \cos{\left(3 p \right)} = 1$$
so
$$\cos{\left(3 p \right)} = 1$$
and
$$\sin{\left(3 p \right)} = 0$$
then
$$p = \frac{2 \pi N}{3}$$
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:
$$z_{1} = \frac{2}{3}$$
$$z_{2} = - \frac{1}{3} - \frac{\sqrt{3} i}{3}$$
$$z_{3} = - \frac{1}{3} + \frac{\sqrt{3} i}{3}$$
do backward replacement
$$z = w$$
$$w = z$$
The final answer:
$$w_{1} = \frac{2}{3}$$
$$w_{2} = - \frac{1}{3} - \frac{\sqrt{3} i}{3}$$
$$w_{3} = - \frac{1}{3} + \frac{\sqrt{3} i}{3}$$
do backward replacement
$$\cos{\left(x \right)} = w$$
Given the equation
$$\cos{\left(x \right)} = w$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = \pi n + \operatorname{acos}{\left(w \right)}$$
$$x = \pi n + \operatorname{acos}{\left(w \right)} - \pi$$
Or
$$x = \pi n + \operatorname{acos}{\left(w \right)}$$
$$x = \pi n + \operatorname{acos}{\left(w \right)} - \pi$$
, where n - is a integer
substitute w:
$$x_{1} = \pi n + \operatorname{acos}{\left(w_{1} \right)}$$
$$x_{1} = \pi n + \operatorname{acos}{\left(\frac{2}{3} \right)}$$
$$x_{1} = \pi n + \operatorname{acos}{\left(\frac{2}{3} \right)}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(w_{1} \right)} - \pi$$
$$x_{2} = \pi n - \pi + \operatorname{acos}{\left(\frac{2}{3} \right)}$$
$$x_{2} = \pi n - \pi + \operatorname{acos}{\left(\frac{2}{3} \right)}$$
$$27 \cos^{3}{\left(x \right)} - 8 = 0$$
transform
$$27 \cos^{3}{\left(x \right)} - 8 = 0$$
$$27 \cos^{3}{\left(x \right)} - 8 = 0$$
Do replacement
$$w = \cos{\left(x \right)}$$
Given the equation
$$27 w^{3} - 8 = 0$$
Because equation degree is equal to = 3 - does not contain even numbers in the numerator, then
the equation has single real root.
Get the root 3-th degree of the equation sides:
We get:
$$\sqrt[3]{27} \sqrt[3]{w^{3}} = \sqrt[3]{8}$$
or
$$3 w = 2$$
Divide both parts of the equation by 3
w = 2 / (3)
We get the answer: w = 2/3
All other 2 root(s) is the complex numbers.
do replacement:
$$z = w$$
then the equation will be the:
$$z^{3} = \frac{8}{27}$$
Any complex number can presented so:
$$z = r e^{i p}$$
substitute to the equation
$$r^{3} e^{3 i p} = \frac{8}{27}$$
where
$$r = \frac{2}{3}$$
- the magnitude of the complex number
Substitute r:
$$e^{3 i p} = 1$$
Using Euler’s formula, we find roots for p
$$i \sin{\left(3 p \right)} + \cos{\left(3 p \right)} = 1$$
so
$$\cos{\left(3 p \right)} = 1$$
and
$$\sin{\left(3 p \right)} = 0$$
then
$$p = \frac{2 \pi N}{3}$$
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:
$$z_{1} = \frac{2}{3}$$
$$z_{2} = - \frac{1}{3} - \frac{\sqrt{3} i}{3}$$
$$z_{3} = - \frac{1}{3} + \frac{\sqrt{3} i}{3}$$
do backward replacement
$$z = w$$
$$w = z$$
The final answer:
$$w_{1} = \frac{2}{3}$$
$$w_{2} = - \frac{1}{3} - \frac{\sqrt{3} i}{3}$$
$$w_{3} = - \frac{1}{3} + \frac{\sqrt{3} i}{3}$$
do backward replacement
$$\cos{\left(x \right)} = w$$
Given the equation
$$\cos{\left(x \right)} = w$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = \pi n + \operatorname{acos}{\left(w \right)}$$
$$x = \pi n + \operatorname{acos}{\left(w \right)} - \pi$$
Or
$$x = \pi n + \operatorname{acos}{\left(w \right)}$$
$$x = \pi n + \operatorname{acos}{\left(w \right)} - \pi$$
, where n - is a integer
substitute w:
$$x_{1} = \pi n + \operatorname{acos}{\left(w_{1} \right)}$$
$$x_{1} = \pi n + \operatorname{acos}{\left(\frac{2}{3} \right)}$$
$$x_{1} = \pi n + \operatorname{acos}{\left(\frac{2}{3} \right)}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(w_{1} \right)} - \pi$$
$$x_{2} = \pi n - \pi + \operatorname{acos}{\left(\frac{2}{3} \right)}$$
$$x_{2} = \pi n - \pi + \operatorname{acos}{\left(\frac{2}{3} \right)}$$
Rapid solution
[src]
x1 = -acos(2/3) + 2*pi
$$x_{1} = - \operatorname{acos}{\left(\frac{2}{3} \right)} + 2 \pi$$
x2 = acos(2/3)
$$x_{2} = \operatorname{acos}{\left(\frac{2}{3} \right)}$$
/ / ___\\ / / ___\\
| | 1 I*\/ 3 || | | 1 I*\/ 3 ||
x3 = - re|acos|- - - -------|| + 2*pi - I*im|acos|- - - -------||
\ \ 3 3 // \ \ 3 3 //
$$x_{3} = - \operatorname{re}{\left(\operatorname{acos}{\left(- \frac{1}{3} - \frac{\sqrt{3} i}{3} \right)}\right)} + 2 \pi - i \operatorname{im}{\left(\operatorname{acos}{\left(- \frac{1}{3} - \frac{\sqrt{3} i}{3} \right)}\right)}$$
/ / ___\\ / / ___\\
| | 1 I*\/ 3 || | | 1 I*\/ 3 ||
x4 = - re|acos|- - + -------|| + 2*pi - I*im|acos|- - + -------||
\ \ 3 3 // \ \ 3 3 //
$$x_{4} = - \operatorname{re}{\left(\operatorname{acos}{\left(- \frac{1}{3} + \frac{\sqrt{3} i}{3} \right)}\right)} + 2 \pi - i \operatorname{im}{\left(\operatorname{acos}{\left(- \frac{1}{3} + \frac{\sqrt{3} i}{3} \right)}\right)}$$
/ / ___\\ / / ___\\
| | 1 I*\/ 3 || | | 1 I*\/ 3 ||
x5 = I*im|acos|- - - -------|| + re|acos|- - - -------||
\ \ 3 3 // \ \ 3 3 //
$$x_{5} = \operatorname{re}{\left(\operatorname{acos}{\left(- \frac{1}{3} - \frac{\sqrt{3} i}{3} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(- \frac{1}{3} - \frac{\sqrt{3} i}{3} \right)}\right)}$$
/ / ___\\ / / ___\\
| | 1 I*\/ 3 || | | 1 I*\/ 3 ||
x6 = I*im|acos|- - + -------|| + re|acos|- - + -------||
\ \ 3 3 // \ \ 3 3 //
$$x_{6} = \operatorname{re}{\left(\operatorname{acos}{\left(- \frac{1}{3} + \frac{\sqrt{3} i}{3} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(- \frac{1}{3} + \frac{\sqrt{3} i}{3} \right)}\right)}$$
x6 = re(acos(-1/3 + sqrt(3)*i/3)) + i*im(acos(-1/3 + sqrt(3)*i/3))
Sum and product of roots
[src]
sum
/ / ___\\ / / ___\\ / / ___\\ / / ___\\ / / ___\\ / / ___\\ / / ___\\ / / ___\\
| | 1 I*\/ 3 || | | 1 I*\/ 3 || | | 1 I*\/ 3 || | | 1 I*\/ 3 || | | 1 I*\/ 3 || | | 1 I*\/ 3 || | | 1 I*\/ 3 || | | 1 I*\/ 3 ||
-acos(2/3) + 2*pi + acos(2/3) + - re|acos|- - - -------|| + 2*pi - I*im|acos|- - - -------|| + - re|acos|- - + -------|| + 2*pi - I*im|acos|- - + -------|| + I*im|acos|- - - -------|| + re|acos|- - - -------|| + I*im|acos|- - + -------|| + re|acos|- - + -------||
\ \ 3 3 // \ \ 3 3 // \ \ 3 3 // \ \ 3 3 // \ \ 3 3 // \ \ 3 3 // \ \ 3 3 // \ \ 3 3 //
$$\left(\operatorname{re}{\left(\operatorname{acos}{\left(- \frac{1}{3} + \frac{\sqrt{3} i}{3} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(- \frac{1}{3} + \frac{\sqrt{3} i}{3} \right)}\right)}\right) + \left(\left(\left(\left(\operatorname{acos}{\left(\frac{2}{3} \right)} + \left(- \operatorname{acos}{\left(\frac{2}{3} \right)} + 2 \pi\right)\right) + \left(- \operatorname{re}{\left(\operatorname{acos}{\left(- \frac{1}{3} - \frac{\sqrt{3} i}{3} \right)}\right)} + 2 \pi - i \operatorname{im}{\left(\operatorname{acos}{\left(- \frac{1}{3} - \frac{\sqrt{3} i}{3} \right)}\right)}\right)\right) + \left(- \operatorname{re}{\left(\operatorname{acos}{\left(- \frac{1}{3} + \frac{\sqrt{3} i}{3} \right)}\right)} + 2 \pi - i \operatorname{im}{\left(\operatorname{acos}{\left(- \frac{1}{3} + \frac{\sqrt{3} i}{3} \right)}\right)}\right)\right) + \left(\operatorname{re}{\left(\operatorname{acos}{\left(- \frac{1}{3} - \frac{\sqrt{3} i}{3} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(- \frac{1}{3} - \frac{\sqrt{3} i}{3} \right)}\right)}\right)\right)$$
=
6*pi
$$6 \pi$$
product
/ / / ___\\ / / ___\\\ / / / ___\\ / / ___\\\ / / / ___\\ / / ___\\\ / / / ___\\ / / ___\\\
| | | 1 I*\/ 3 || | | 1 I*\/ 3 ||| | | | 1 I*\/ 3 || | | 1 I*\/ 3 ||| | | | 1 I*\/ 3 || | | 1 I*\/ 3 ||| | | | 1 I*\/ 3 || | | 1 I*\/ 3 |||
(-acos(2/3) + 2*pi)*acos(2/3)*|- re|acos|- - - -------|| + 2*pi - I*im|acos|- - - -------|||*|- re|acos|- - + -------|| + 2*pi - I*im|acos|- - + -------|||*|I*im|acos|- - - -------|| + re|acos|- - - -------|||*|I*im|acos|- - + -------|| + re|acos|- - + -------|||
\ \ \ 3 3 // \ \ 3 3 /// \ \ \ 3 3 // \ \ 3 3 /// \ \ \ 3 3 // \ \ 3 3 /// \ \ \ 3 3 // \ \ 3 3 ///
$$\left(- \operatorname{acos}{\left(\frac{2}{3} \right)} + 2 \pi\right) \operatorname{acos}{\left(\frac{2}{3} \right)} \left(- \operatorname{re}{\left(\operatorname{acos}{\left(- \frac{1}{3} - \frac{\sqrt{3} i}{3} \right)}\right)} + 2 \pi - i \operatorname{im}{\left(\operatorname{acos}{\left(- \frac{1}{3} - \frac{\sqrt{3} i}{3} \right)}\right)}\right) \left(- \operatorname{re}{\left(\operatorname{acos}{\left(- \frac{1}{3} + \frac{\sqrt{3} i}{3} \right)}\right)} + 2 \pi - i \operatorname{im}{\left(\operatorname{acos}{\left(- \frac{1}{3} + \frac{\sqrt{3} i}{3} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{acos}{\left(- \frac{1}{3} - \frac{\sqrt{3} i}{3} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(- \frac{1}{3} - \frac{\sqrt{3} i}{3} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{acos}{\left(- \frac{1}{3} + \frac{\sqrt{3} i}{3} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(- \frac{1}{3} + \frac{\sqrt{3} i}{3} \right)}\right)}\right)$$
=
/ / / ___\\ / / ___\\\ / / / ___\\ / / ___\\\ / / / ___\\ / / ___\\\ / / / ___\\ / / ___\\\
| | | 1 I*\/ 3 || | | 1 I*\/ 3 ||| | | | 1 I*\/ 3 || | | 1 I*\/ 3 ||| | | | 1 I*\/ 3 || | | 1 I*\/ 3 ||| | | | 1 I*\/ 3 || | | 1 I*\/ 3 |||
(-acos(2/3) + 2*pi)*|I*im|acos|- - - -------|| + re|acos|- - - -------|||*|I*im|acos|- - + -------|| + re|acos|- - + -------|||*|-2*pi + I*im|acos|- - - -------|| + re|acos|- - - -------|||*|-2*pi + I*im|acos|- - + -------|| + re|acos|- - + -------|||*acos(2/3)
\ \ \ 3 3 // \ \ 3 3 /// \ \ \ 3 3 // \ \ 3 3 /// \ \ \ 3 3 // \ \ 3 3 /// \ \ \ 3 3 // \ \ 3 3 ///
$$\left(\operatorname{re}{\left(\operatorname{acos}{\left(- \frac{1}{3} - \frac{\sqrt{3} i}{3} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(- \frac{1}{3} - \frac{\sqrt{3} i}{3} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{acos}{\left(- \frac{1}{3} + \frac{\sqrt{3} i}{3} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(- \frac{1}{3} + \frac{\sqrt{3} i}{3} \right)}\right)}\right) \left(- \operatorname{acos}{\left(\frac{2}{3} \right)} + 2 \pi\right) \left(- 2 \pi + \operatorname{re}{\left(\operatorname{acos}{\left(- \frac{1}{3} - \frac{\sqrt{3} i}{3} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(- \frac{1}{3} - \frac{\sqrt{3} i}{3} \right)}\right)}\right) \left(- 2 \pi + \operatorname{re}{\left(\operatorname{acos}{\left(- \frac{1}{3} + \frac{\sqrt{3} i}{3} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(- \frac{1}{3} + \frac{\sqrt{3} i}{3} \right)}\right)}\right) \operatorname{acos}{\left(\frac{2}{3} \right)}$$
(-acos(2/3) + 2*pi)*(i*im(acos(-1/3 - i*sqrt(3)/3)) + re(acos(-1/3 - i*sqrt(3)/3)))*(i*im(acos(-1/3 + i*sqrt(3)/3)) + re(acos(-1/3 + i*sqrt(3)/3)))*(-2*pi + i*im(acos(-1/3 - i*sqrt(3)/3)) + re(acos(-1/3 - i*sqrt(3)/3)))*(-2*pi + i*im(acos(-1/3 + i*sqrt(3)/3)) + re(acos(-1/3 + i*sqrt(3)/3)))*acos(2/3)
Numerical answer
[src]
x1 = 105.973081551485
x2 = 82.5224776639025
x3 = 32.2569952064659
x4 = -36.8580431725096
x5 = 214.469369114674
x6 = 7.12425397774752
x7 = -57.3897364351842
x8 = -24.2916725581504
x9 = 74.5571550155871
x10 = 61.9907844012279
x11 = -99.6898962443055
x12 = 101.372033585441
x13 = 43.1412284796892
x14 = -80.8403403227667
x15 = 30.57485786533
x16 = -82.5224776639025
x17 = 24.2916725581504
x18 = -13.4074392849271
x19 = -63.6729217423638
x20 = -76.239292356723
x21 = 11.7253019437912
x22 = -32.2569952064659
x23 = -25.9738098992863
x24 = -61.9907844012279
x25 = -49.4244137868688
x26 = 124.822637473024
x27 = -88.8056629710821
x28 = 0.84106867056793
x29 = 99.6898962443055
x30 = 68.2739697084075
x31 = -69.9561070495434
x32 = 63.6729217423638
x33 = -68.2739697084075
x34 = -38.5401805136454
x35 = 38.5401805136454
x36 = -11.7253019437912
x37 = -93.4067109371259
x38 = 44.823365820825
x39 = 95.0888482782617
x40 = -5.44211663661166
x41 = 55.7075990940483
x42 = -44.823365820825
x43 = -0.84106867056793
x44 = 49.4244137868688
x45 = 76.239292356723
x46 = -55.7075990940483
x47 = -101.372033585441
x48 = 19.6906245921067
x49 = 88.8056629710821
x50 = -18.0084872509708
x51 = -87.1235256299463
x52 = 51.1065511280046
x53 = 87.1235256299463
x54 = 5.44211663661166
x55 = -19.6906245921067
x56 = 18.0084872509708
x57 = -43.1412284796892
x58 = 69.9561070495434
x59 = 25.9738098992863
x59 = 25.9738098992863