Mister Exam
Other calculators
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How to use it?
- Equation:
-
Equation cos(x)=√3/2
- Equation x^2-x-56=0
-
Equation cos(x)=-1/4
-
Equation |a|=9
- Express {x} in terms of y where:
- -19*x-4*y=5
- 19*x+3*y=-3
- -6*x-7*y=18
- 10*x-9*y=-1
- Sum of series:
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-1
- Derivative of:
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-1
- Integral of d{x}:
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-1
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Identical expressions
- log2cos(x)=- one
- logarithm of 2 co sinus of e of (x) equally minus 1
- logarithm of 2 co sinus of e of (x) equally minus one
- log2cosx=-1
-
Similar expressions
- log2cos(x)=+1
- log(2cosx)=2
- 2(log(2cosx)/log(3))−13(log(2cosx)/log(3))+6=0
- log(2cosx)/log3=4
- log2cosx=-1
log2cos(x)=-1 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation
$$\log{\left(2 \cos{\left(x \right)} \right)} = -1$$
transform
$$\log{\left(2 \cos{\left(x \right)} \right)} + 1 = 0$$
$$\log{\left(2 \cos{\left(x \right)} \right)} + 1 = 0$$
Do replacement
$$w = \cos{\left(x \right)}$$
Given the equation
$$\log{\left(2 w \right)} + 1 = 0$$
$$\log{\left(2 w \right)} = -1$$
This equation is of the form:
By definition log
then
$$2 w = e^{- 1^{-1}}$$
simplify
$$2 w = e^{-1}$$
$$w = \frac{1}{2 e}$$
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:
$$\log{\left(2 \cos{\left(x \right)} \right)} = -1$$
transform
$$\log{\left(2 \cos{\left(x \right)} \right)} + 1 = 0$$
$$\log{\left(2 \cos{\left(x \right)} \right)} + 1 = 0$$
Do replacement
$$w = \cos{\left(x \right)}$$
Given the equation
$$\log{\left(2 w \right)} + 1 = 0$$
$$\log{\left(2 w \right)} = -1$$
This equation is of the form:
log(v)=p
By definition log
v=e^p
then
$$2 w = e^{- 1^{-1}}$$
simplify
$$2 w = e^{-1}$$
$$w = \frac{1}{2 e}$$
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\ / -1\
|e | |e |
- acos|---| + 2*pi + acos|---|
\ 2 / \ 2 /
$$\operatorname{acos}{\left(\frac{1}{2 e} \right)} + \left(- \operatorname{acos}{\left(\frac{1}{2 e} \right)} + 2 \pi\right)$$
=
2*pi
$$2 \pi$$
product
/ / -1\ \ / -1\ | |e | | |e | |- acos|---| + 2*pi|*acos|---| \ \ 2 / / \ 2 /
$$\left(- \operatorname{acos}{\left(\frac{1}{2 e} \right)} + 2 \pi\right) \operatorname{acos}{\left(\frac{1}{2 e} \right)}$$
=
/ / -1\ \ / -1\ | |e | | |e | |- acos|---| + 2*pi|*acos|---| \ \ 2 / / \ 2 /
$$\left(- \operatorname{acos}{\left(\frac{1}{2 e} \right)} + 2 \pi\right) \operatorname{acos}{\left(\frac{1}{2 e} \right)}$$
(-acos(exp(-1)/2) + 2*pi)*acos(exp(-1)/2)
Rapid solution
[src]
/ -1\
|e |
x1 = - acos|---| + 2*pi
\ 2 /
$$x_{1} = - \operatorname{acos}{\left(\frac{1}{2 e} \right)} + 2 \pi$$
/ -1\
|e |
x2 = acos|---|
\ 2 /
$$x_{2} = \operatorname{acos}{\left(\frac{1}{2 e} \right)}$$
x2 = acos(exp(-1)/2)
Numerical answer
[src]
x1 = 20.235359179544
x2 = -20.235359179544
x3 = 74.0124204281498
x4 = -61.4460498137906
x5 = -51.6512857154419
x6 = 30.0301232778927
x7 = -7.66898856518482
x8 = -13.9521738723644
x9 = -64.2176563298011
x10 = -67.7292351209702
x11 = -57.9344710226215
x12 = -74.0124204281498
x13 = 26.5185444867236
x14 = -23.7469379707131
x15 = 70.5008416369807
x16 = -17.4637526635335
x17 = 13.9521738723644
x18 = 57.9344710226215
x19 = -95.633582865699
x20 = -30.0301232778927
x21 = 67.7292351209702
x22 = 36.3133085850723
x23 = 23.7469379707131
x24 = 64.2176563298011
x25 = 80.2956057353294
x25 = 80.2956057353294