Mister Exam
Other calculators
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How to use it?
- Equation:
-
Equation -9*(8-9x)=4x+5
-
Equation 20-2*x=27+x
-
Equation sin(x)+cos(x)=0
-
Equation sin(x)-cos(x)=0
- Express {x} in terms of y where:
- (x^2+y^2)×(x+y-3)=2xy
- ((42^2-22)+(67*y))/67=((21^3-22)+(67*x))/67
- sqrt(1-x)^2+(4-y)^2=0
- lnx+sin(y/x+2x)=4
-
Identical expressions
- log(two cosx)=2
- logarithm of (2 co sinus of e of x) equally 2
- logarithm of (two co sinus of e of x) equally 2
- log2cosx=2
-
Similar expressions
- log2cos(x)=0.2
log(2cosx)=2 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation
$$\log{\left(2 \cos{\left(x \right)} \right)} = 2$$
transform
$$\log{\left(2 \cos{\left(x \right)} \right)} - 2 = 0$$
$$\log{\left(2 \cos{\left(x \right)} \right)} - 2 = 0$$
Do replacement
$$w = \cos{\left(x \right)}$$
Given the equation
$$\log{\left(2 w \right)} - 2 = 0$$
$$\log{\left(2 w \right)} = 2$$
This equation is of the form:
By definition log
then
$$2 w = e^{\frac{2}{1}}$$
simplify
$$2 w = e^{2}$$
$$w = \frac{e^{2}}{2}$$
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:
$$\log{\left(2 \cos{\left(x \right)} \right)} = 2$$
transform
$$\log{\left(2 \cos{\left(x \right)} \right)} - 2 = 0$$
$$\log{\left(2 \cos{\left(x \right)} \right)} - 2 = 0$$
Do replacement
$$w = \cos{\left(x \right)}$$
Given the equation
$$\log{\left(2 w \right)} - 2 = 0$$
$$\log{\left(2 w \right)} = 2$$
This equation is of the form:
log(v)=p
By definition log
v=e^p
then
$$2 w = e^{\frac{2}{1}}$$
simplify
$$2 w = e^{2}$$
$$w = \frac{e^{2}}{2}$$
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:
Sum and product of roots
[src]
sum
/ / 2\\ / / 2\\ / / 2\\
| |e || | |e || | |e ||
2*pi - I*im|acos|--|| + I*im|acos|--|| + re|acos|--||
\ \2 // \ \2 // \ \2 //
$$\left(2 \pi - i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{e^{2}}{2} \right)}\right)}\right) + \left(\operatorname{re}{\left(\operatorname{acos}{\left(\frac{e^{2}}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{e^{2}}{2} \right)}\right)}\right)$$
=
/ / 2\\
| |e ||
2*pi + re|acos|--||
\ \2 //
$$\operatorname{re}{\left(\operatorname{acos}{\left(\frac{e^{2}}{2} \right)}\right)} + 2 \pi$$
product
/ / / 2\\\ / / / 2\\ / / 2\\\ | | |e ||| | | |e || | |e ||| |2*pi - I*im|acos|--|||*|I*im|acos|--|| + re|acos|--||| \ \ \2 /// \ \ \2 // \ \2 ///
$$\left(2 \pi - i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{e^{2}}{2} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{acos}{\left(\frac{e^{2}}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{e^{2}}{2} \right)}\right)}\right)$$
=
/ / / 2\\\ / / / 2\\ / / 2\\\ | | |e ||| | | |e || | |e ||| |2*pi - I*im|acos|--|||*|I*im|acos|--|| + re|acos|--||| \ \ \2 /// \ \ \2 // \ \2 ///
$$\left(2 \pi - i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{e^{2}}{2} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{acos}{\left(\frac{e^{2}}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{e^{2}}{2} \right)}\right)}\right)$$
(2*pi - i*im(acos(exp(2)/2)))*(i*im(acos(exp(2)/2)) + re(acos(exp(2)/2)))
Rapid solution
[src]
/ / 2\\
| |e ||
x1 = 2*pi - I*im|acos|--||
\ \2 //
$$x_{1} = 2 \pi - i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{e^{2}}{2} \right)}\right)}$$
/ / 2\\ / / 2\\
| |e || | |e ||
x2 = I*im|acos|--|| + re|acos|--||
\ \2 // \ \2 //
$$x_{2} = \operatorname{re}{\left(\operatorname{acos}{\left(\frac{e^{2}}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{e^{2}}{2} \right)}\right)}$$
x2 = re(acos(exp(2)/2)) + i*im(acos(exp(2)/2))
Numerical answer
[src]
x1 = 12.5663706143592 - 1.98115964675051*i
x2 = 87.9645943005142 - 1.98115964675051*i
x3 = -94.2477796076938 + 1.98115964675051*i
x4 = -251.327412287183 - 1.98115964675051*i
x5 = -18.8495559215388 + 1.98115964675051*i
x6 = 37.6991118430775 + 1.98115964675051*i
x7 = -81.6814089933346 - 1.98115964675051*i
x8 = -75.398223686155 - 1.98115964675051*i
x9 = 87.9645943005142 + 1.98115964675051*i
x10 = -6.28318530717959 - 1.98115964675051*i
x11 = 18.8495559215388 + 1.98115964675051*i
x12 = 9003.80454518835 - 1.98115964675051*i
x13 = 50.2654824574367 - 1.98115964675051*i
x14 = -50.2654824574367 + 1.98115964675051*i
x15 = 94.2477796076938 - 1.98115964675051*i
x16 = 1.24469990737641e-32 + 1.98115964675051*i
x17 = -31.4159265358979 - 1.98115964675051*i
x18 = -50.2654824574367 - 1.98115964675051*i
x19 = -69.1150383789755 + 1.98115964675051*i
x20 = -62.8318530717959 + 1.98115964675051*i
x21 = -18.8495559215388 - 1.98115964675051*i
x22 = 131.946891450771 + 1.98115964675051*i
x23 = 31.4159265358979 - 1.98115964675051*i
x24 = 94.2477796076938 + 1.98115964675051*i
x25 = 6.28318530717959 - 1.98115964675051*i
x26 = 50.2654824574367 + 1.98115964675051*i
x27 = -43.9822971502571 - 1.98115964675051*i
x28 = -62.8318530717959 - 1.98115964675051*i
x29 = -87.9645943005142 + 1.98115964675051*i
x30 = 1.84368999347033e-35 + 1.98115964675051*i
x31 = -6.28318530717959 + 1.98115964675051*i
x32 = 69.1150383789755 - 1.98115964675051*i
x33 = -37.6991118430775 - 1.98115964675051*i
x34 = 43.9822971502571 - 1.98115964675051*i
x35 = -81.6814089933346 + 1.98115964675051*i
x36 = -12.5663706143592 - 1.98115964675051*i
x37 = 1.98115964675051*i
x38 = 43.9822971502571 + 1.98115964675051*i
x39 = 5.93472984109987e-67 - 1.98115964675051*i
x40 = -100.530964914873 - 1.98115964675051*i
x41 = 100.530964914873 - 1.98115964675051*i
x42 = 69.1150383789755 + 1.98115964675051*i
x43 = -87.9645943005142 - 1.98115964675051*i
x44 = -282.743338823081 + 1.98115964675051*i
x45 = -43.9822971502571 + 1.98115964675051*i
x46 = -56.5486677646163 - 1.98115964675051*i
x47 = 75.398223686155 - 1.98115964675051*i
x48 = -25.1327412287183 + 1.98115964675051*i
x49 = 81.6814089933346 + 1.98115964675051*i
x50 = 622.035345410779 + 1.98115964675051*i
x51 = 6.28318530717959 + 1.98115964675051*i
x52 = 56.5486677646163 - 1.98115964675051*i
x53 = -364.424747816416 - 1.98115964675051*i
x54 = 25.1327412287183 + 1.98115964675051*i
x55 = -37.6991118430775 + 1.98115964675051*i
x56 = 25.1327412287183 - 1.98115964675051*i
x57 = 62.8318530717959 + 1.98115964675051*i
x57 = 62.8318530717959 + 1.98115964675051*i