Mister Exam
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- Equation:
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Equation 4^(x-1/2)-6*2^(x-1)+3=0
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Equation sin(x+1)=0
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Equation 2*x/(x^2+1)=0
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- Integral of d{x}:
- 0.2
- Sum of series:
-
0.2
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Identical expressions
- log two cos(x)= zero .2
- logarithm of 2 co sinus of e of (x) equally 0.2
- logarithm of two co sinus of e of (x) equally zero .2
- log2cosx=0.2
- log2cos(x)=O.2
-
Similar expressions
- log(2cosx)/log3=4
- log2cosx=0.2
log2cos(x)=0.2 equation
The teacher will be very surprised to see your correct solution 😉
The solution
You have entered
[src]
log(2*cos(x)) = 1/5
$$\log{\left(2 \cos{\left(x \right)} \right)} = \frac{1}{5}$$
Detail solution
Given the equation
$$\log{\left(2 \cos{\left(x \right)} \right)} = \frac{1}{5}$$
transform
$$\log{\left(2 \cos{\left(x \right)} \right)} - \frac{1}{5} = 0$$
$$\log{\left(2 \cos{\left(x \right)} \right)} - \frac{1}{5} = 0$$
Do replacement
$$w = \cos{\left(x \right)}$$
Given the equation
$$\log{\left(2 w \right)} - \frac{1}{5} = 0$$
$$\log{\left(2 w \right)} = \frac{1}{5}$$
This equation is of the form:
By definition log
then
$$2 w = e^{\frac{1}{5}}$$
simplify
$$2 w = e^{\frac{1}{5}}$$
$$w = \frac{e^{\frac{1}{5}}}{2}$$
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:
$$\log{\left(2 \cos{\left(x \right)} \right)} = \frac{1}{5}$$
transform
$$\log{\left(2 \cos{\left(x \right)} \right)} - \frac{1}{5} = 0$$
$$\log{\left(2 \cos{\left(x \right)} \right)} - \frac{1}{5} = 0$$
Do replacement
$$w = \cos{\left(x \right)}$$
Given the equation
$$\log{\left(2 w \right)} - \frac{1}{5} = 0$$
$$\log{\left(2 w \right)} = \frac{1}{5}$$
This equation is of the form:
log(v)=p
By definition log
v=e^p
then
$$2 w = e^{\frac{1}{5}}$$
simplify
$$2 w = e^{\frac{1}{5}}$$
$$w = \frac{e^{\frac{1}{5}}}{2}$$
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
/ 1/5\ / 1/5\
|e | |e |
- acos|----| + 2*pi + acos|----|
\ 2 / \ 2 /
$$\operatorname{acos}{\left(\frac{e^{\frac{1}{5}}}{2} \right)} + \left(- \operatorname{acos}{\left(\frac{e^{\frac{1}{5}}}{2} \right)} + 2 \pi\right)$$
=
2*pi
$$2 \pi$$
product
/ / 1/5\ \ / 1/5\ | |e | | |e | |- acos|----| + 2*pi|*acos|----| \ \ 2 / / \ 2 /
$$\left(- \operatorname{acos}{\left(\frac{e^{\frac{1}{5}}}{2} \right)} + 2 \pi\right) \operatorname{acos}{\left(\frac{e^{\frac{1}{5}}}{2} \right)}$$
=
/ / 1/5\ \ / 1/5\ | |e | | |e | |- acos|----| + 2*pi|*acos|----| \ \ 2 / / \ 2 /
$$\left(- \operatorname{acos}{\left(\frac{e^{\frac{1}{5}}}{2} \right)} + 2 \pi\right) \operatorname{acos}{\left(\frac{e^{\frac{1}{5}}}{2} \right)}$$
(-acos(exp(1/5)/2) + 2*pi)*acos(exp(1/5)/2)
Rapid solution
[src]
/ 1/5\
|e |
x1 = - acos|----| + 2*pi
\ 2 /
$$x_{1} = - \operatorname{acos}{\left(\frac{e^{\frac{1}{5}}}{2} \right)} + 2 \pi$$
/ 1/5\
|e |
x2 = acos|----|
\ 2 /
$$x_{2} = \operatorname{acos}{\left(\frac{e^{\frac{1}{5}}}{2} \right)}$$
x2 = acos(exp(1/5)/2)
Numerical answer
[src]
x1 = -51.179332760397
x2 = -36.7852615401172
x3 = 30.5020762329377
x4 = 7.19703561013987
x5 = 0.913850302960281
x6 = -17.9357056185785
x7 = 49.3516321544764
x8 = 82.5952592962949
x9 = -74.4843733831948
x10 = 63.7457033747561
x11 = 80.7675586903743
x12 = 43.0684468472968
x13 = 101.444815217834
x14 = -61.9180027688356
x15 = 44.8961474532174
x16 = 87.0507439975539
x17 = -82.5952592962949
x18 = -7.19703561013987
x19 = 19.763406224499
x20 = -80.7675586903743
x21 = -13.4802209173195
x22 = -49.3516321544764
x23 = -57.4625180675766
x24 = -95.1616299106541
x25 = 93.3339293047335
x26 = 68.2011880760152
x27 = -101.444815217834
x28 = 38.6129621460378
x29 = 5.36933500421931
x30 = 88.8784446034745
x31 = -5.36933500421931
x32 = 57.4625180675766
x33 = 32.3297768388582
x34 = 74.4843733831948
x35 = 26.0465915316786
x36 = -68.2011880760152
x37 = 55.634817461656
x38 = -38.6129621460378
x39 = -70.0288886819357
x40 = -99.6171146119131
x41 = -24.2188909257581
x42 = -30.5020762329377
x43 = 70.0288886819357
x44 = 24.2188909257581
x45 = 11.6525203113989
x46 = -87.0507439975539
x47 = -76.3120739891153
x48 = -0.913850302960281
x49 = 17.9357056185785
x50 = 76.3120739891153
x51 = -19.763406224499
x52 = -44.8961474532174
x53 = -55.634817461656
x54 = -11.6525203113989
x55 = 13.4802209173195
x56 = 36.7852615401172
x57 = 99.6171146119131
x58 = -63.7457033747561
x59 = -93.3339293047335
x60 = -43.0684468472968
x61 = -26.0465915316786
x62 = -32.3297768388582
x63 = 61.9180027688356
x63 = 61.9180027688356