Mister Exam
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- Equation:
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Equation (x+1)^2+(x-6)^2=2x^2
- Equation x³-2x²+x=0
- Equation sin3x+sinx-sin2x=2cos^2x-2cosx
- Equation x^(log(x,1/3)+4)=1/243
- Express {x} in terms of y where:
- -1*x-19*y=-10
- 5*x+5*y=20
- 4*x-16*y=6
- sqrt(x^2+y^2+4x-6y-12)+(x^2+10x-27)=8-|y-9|+|y-3|
- Derivative of:
- -1/2
- Limit of the function:
- -1/2
- Integral of d{x}:
-
-1/2
-
Identical expressions
- log(three)(-cosx)+log one / three (-sinx)=-1/ two
- logarithm of (3)( minus co sinus of e of x) plus logarithm of 1 divide by 3( minus sinus of x) equally minus 1 divide by 2
- logarithm of (three)( minus co sinus of e of x) plus logarithm of one divide by three ( minus sinus of x) equally minus 1 divide by two
- log3-cosx+log1/3-sinx=-1/2
- log(3)(-cosx)+log1 divide by 3(-sinx)=-1 divide by 2
-
Similar expressions
- log(3)(-cosx)+log1/3(sinx)=-1/2
- cos^23x-sin^23x=-(1/2)
- log(3)(-cosx)+log1/3(-sinx)=+1/2
- log(3)(cosx)+log1/3(-sinx)=-1/2
- log(3)(-cosx)-log1/3(-sinx)=-1/2
- sqrt((2-x)/(x-1))-7(sqrt((x-1)/(2-x)))=6
log(3)(-cosx)+log1/3(-sinx)=-1/2 equation
The teacher will be very surprised to see your correct solution 😉
The solution
You have entered
[src]
log(1)*-sin(x)
log(3)*-cos(x) + -------------- = -1/2
3
$$\frac{\log{\left(1 \right)} \left(- \sin{\left(x \right)}\right)}{3} + \log{\left(3 \right)} \left(- \cos{\left(x \right)}\right) = - \frac{1}{2}$$
Detail solution
Given the equation
$$\frac{\log{\left(1 \right)} \left(- \sin{\left(x \right)}\right)}{3} + \log{\left(3 \right)} \left(- \cos{\left(x \right)}\right) = - \frac{1}{2}$$
- this is the simplest trigonometric equation
Divide both parts of the equation by $- \log{\left(3 \right)}$
The equation is transformed to
$$\cos{\left(x \right)} = \frac{1}{2 \log{\left(3 \right)}}$$
This equation is transformed to
$$x = 2 \pi n + \operatorname{acos}{\left(\frac{1}{2 \log{\left(3 \right)}} \right)}$$
$$x = 2 \pi n - \pi + \operatorname{acos}{\left(\frac{1}{2 \log{\left(3 \right)}} \right)}$$
Or
$$x = 2 \pi n + \operatorname{acos}{\left(\frac{1}{2 \log{\left(3 \right)}} \right)}$$
$$x = 2 \pi n - \pi + \operatorname{acos}{\left(\frac{1}{2 \log{\left(3 \right)}} \right)}$$
, where n - is a integer
$$\frac{\log{\left(1 \right)} \left(- \sin{\left(x \right)}\right)}{3} + \log{\left(3 \right)} \left(- \cos{\left(x \right)}\right) = - \frac{1}{2}$$
- this is the simplest trigonometric equation
Divide both parts of the equation by $- \log{\left(3 \right)}$
The equation is transformed to
$$\cos{\left(x \right)} = \frac{1}{2 \log{\left(3 \right)}}$$
This equation is transformed to
$$x = 2 \pi n + \operatorname{acos}{\left(\frac{1}{2 \log{\left(3 \right)}} \right)}$$
$$x = 2 \pi n - \pi + \operatorname{acos}{\left(\frac{1}{2 \log{\left(3 \right)}} \right)}$$
Or
$$x = 2 \pi n + \operatorname{acos}{\left(\frac{1}{2 \log{\left(3 \right)}} \right)}$$
$$x = 2 \pi n - \pi + \operatorname{acos}{\left(\frac{1}{2 \log{\left(3 \right)}} \right)}$$
, where n - is a integer
Sum and product of roots
[src]
sum
/ 1 \ / 1 \
- acos|--------| + 2*pi + acos|--------|
\2*log(3)/ \2*log(3)/
$$\left(- \operatorname{acos}{\left(\frac{1}{2 \log{\left(3 \right)}} \right)} + 2 \pi\right) + \left(\operatorname{acos}{\left(\frac{1}{2 \log{\left(3 \right)}} \right)}\right)$$
=
2*pi
$$2 \pi$$
product
/ 1 \ / 1 \
- acos|--------| + 2*pi * acos|--------|
\2*log(3)/ \2*log(3)/
$$\left(- \operatorname{acos}{\left(\frac{1}{2 \log{\left(3 \right)}} \right)} + 2 \pi\right) * \left(\operatorname{acos}{\left(\frac{1}{2 \log{\left(3 \right)}} \right)}\right)$$
=
/ / 1 \ \ / 1 \ |- acos|--------| + 2*pi|*acos|--------| \ \2*log(3)/ / \2*log(3)/
$$\left(- \operatorname{acos}{\left(\frac{1}{2 \log{\left(3 \right)}} \right)} + 2 \pi\right) \operatorname{acos}{\left(\frac{1}{2 \log{\left(3 \right)}} \right)}$$
Rapid solution
[src]
/ 1 \
x_1 = - acos|--------| + 2*pi
\2*log(3)/
$$x_{1} = - \operatorname{acos}{\left(\frac{1}{2 \log{\left(3 \right)}} \right)} + 2 \pi$$
/ 1 \
x_2 = acos|--------|
\2*log(3)/
$$x_{2} = \operatorname{acos}{\left(\frac{1}{2 \log{\left(3 \right)}} \right)}$$
Numerical answer
[src]
x1 = 57.6469575498607
x2 = 55.4503779793719
x3 = 42.8840073650127
x4 = -45.0805869355015
x5 = 17.7512661362943
x6 = -82.779698778579
x7 = 32.5142163211423
x8 = -93.1494898224494
x9 = -13.6646603996036
x10 = 1368.63610717991
x11 = 89.0628840857586
x12 = -99.432675129629
x13 = -86.8663045152698
x14 = -1.09828978524442
x15 = -74.2999339009106
x16 = 76.4965134713995
x17 = -1182.33712753501
x18 = 11.4680808291148
x19 = -32.5142163211423
x20 = 80.5831192080902
x21 = 36.6008220578331
x22 = -36.6008220578331
x23 = 49.1671926721923
x24 = 30.3176367506535
x25 = 70.2133281642199
x26 = -26.2310310139628
x27 = -42.8840073650127
x28 = -80.5831192080902
x29 = -38.7974016283219
x30 = -55.4503779793719
x31 = -76.4965134713995
x32 = -57.6469575498607
x33 = -17.7512661362943
x34 = -24.0344514434739
x35 = 13.6646603996036
x36 = 5.18489552193517
x37 = -51.3637722426811
x38 = -30.3176367506535
x39 = -19.9478457067832
x40 = -68.016748593731
x41 = 61.7335632865514
x42 = 7.38147509242401
x43 = 93.1494898224494
x44 = 1.09828978524442
x45 = 95.3460693929382
x46 = 51.3637722426811
x47 = 99.432675129629
x48 = -70.2133281642199
x49 = -61.7335632865514
x50 = 82.779698778579
x51 = -95.3460693929382
x52 = -5.18489552193517
x53 = 181.114084122964
x54 = 24.0344514434739
x55 = 26.2310310139628
x56 = 19.9478457067832
x57 = 63.9301428570403
x58 = -63.9301428570403
x59 = -11.4680808291148
x60 = 38.7974016283219
x61 = 45.0805869355015
x62 = 74.2999339009106
x63 = -7.38147509242401
x64 = -49.1671926721923
x65 = 68.016748593731
x66 = -89.0628840857586
x67 = 86.8663045152698
x67 = 86.8663045152698
The graph