Mister Exam
Other calculators
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How to use it?
- Equation:
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Equation x^3-3*x^2-8*x+24=0
-
Equation log(3*x)=3
-
Equation 2x=-3
-
Equation x-1/3=210
- Express {x} in terms of y where:
- -12*x+10*y=2
- 1*x-17*y=10
- 10*x+7*y=-8
- -8*x+11*y=-8
-
Identical expressions
- log2sin2pi/ fifteen
- logarithm of 2 sinus of 2 Pi divide by 15
- logarithm of 2 sinus of 2 Pi divide by fifteen
- log2sin2pi divide by 15
log2sin2pi/15 equation
The teacher will be very surprised to see your correct solution 😉
The solution
You have entered
[src]
log(2*sin(2*p))*I
----------------- = 0
15
$$\frac{i \log{\left(2 \sin{\left(2 p \right)} \right)}}{15} = 0$$
Detail solution
Given the equation
$$\frac{i \log{\left(2 \sin{\left(2 p \right)} \right)}}{15} = 0$$
transform
$$\frac{i \log{\left(2 \sin{\left(2 p \right)} \right)}}{15} = 0$$
$$\frac{i \log{\left(2 \sin{\left(2 p \right)} \right)}}{15} = 0$$
Do replacement
$$w = \sin{\left(2 p \right)}$$
Given the equation
$$\frac{i \log{\left(2 w \right)}}{15} = 0$$
$$\frac{i \log{\left(2 w \right)}}{15} = 0$$
Let's divide both parts of the equation by the multiplier of log =i/15
$$\log{\left(2 w \right)} = 0$$
This equation is of the form:
By definition log
then
$$2 w = e^{\frac{0}{\frac{1}{15} i}}$$
simplify
$$2 w = 1$$
$$w = \frac{1}{2}$$
do backward replacement
$$\sin{\left(2 p \right)} = w$$
Given the equation
$$\sin{\left(2 p \right)} = w$$
- this is the simplest trigonometric equation
This equation is transformed to
$$2 p = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$2 p = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
Or
$$2 p = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$2 p = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
, where n - is a integer
Divide both parts of the equation by
$$2$$
substitute w:
$$\frac{i \log{\left(2 \sin{\left(2 p \right)} \right)}}{15} = 0$$
transform
$$\frac{i \log{\left(2 \sin{\left(2 p \right)} \right)}}{15} = 0$$
$$\frac{i \log{\left(2 \sin{\left(2 p \right)} \right)}}{15} = 0$$
Do replacement
$$w = \sin{\left(2 p \right)}$$
Given the equation
$$\frac{i \log{\left(2 w \right)}}{15} = 0$$
$$\frac{i \log{\left(2 w \right)}}{15} = 0$$
Let's divide both parts of the equation by the multiplier of log =i/15
$$\log{\left(2 w \right)} = 0$$
This equation is of the form:
log(v)=p
By definition log
v=e^p
then
$$2 w = e^{\frac{0}{\frac{1}{15} i}}$$
simplify
$$2 w = 1$$
$$w = \frac{1}{2}$$
do backward replacement
$$\sin{\left(2 p \right)} = w$$
Given the equation
$$\sin{\left(2 p \right)} = w$$
- this is the simplest trigonometric equation
This equation is transformed to
$$2 p = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$2 p = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
Or
$$2 p = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$2 p = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
, where n - is a integer
Divide both parts of the equation by
$$2$$
substitute w:
The graph
Rapid solution
[src]
pi
p1 = --
12
$$p_{1} = \frac{\pi}{12}$$
5*pi
p2 = ----
12
$$p_{2} = \frac{5 \pi}{12}$$
p2 = 5*pi/12
Sum and product of roots
[src]
sum
pi 5*pi -- + ---- 12 12
$$\frac{\pi}{12} + \frac{5 \pi}{12}$$
=
pi -- 2
$$\frac{\pi}{2}$$
product
pi 5*pi --*---- 12 12
$$\frac{\pi}{12} \frac{5 \pi}{12}$$
=
2 5*pi ----- 144
$$\frac{5 \pi^{2}}{144}$$
5*pi^2/144
Numerical answer
[src]
p1 = 3.40339204138894
p2 = 1.30899693899575
p3 = -4.97418836818384
p4 = 10.7337748997651
p5 = -6.02138591938044
p6 = -8.11578102177363
p7 = -9.16297857297023
p8 = 6.54498469497874
p9 = 0.261799387799149
p10 = 4.45058959258554
p11 = -2.87979326579064
p12 = -1.83259571459405
p12 = -1.83259571459405