Mister Exam
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Equation (x-2)^5+(4-x)^5=2
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Identical expressions
- two *cos(x+pi/ three)= five
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- two multiply by co sinus of e of (x plus Pi divide by three) equally five
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Similar expressions
- 2*cos(x-pi/3)=5
2*cos(x+pi/3)=5 equation
The teacher will be very surprised to see your correct solution 😉
The solution
You have entered
[src]
/ pi\
2*cos|x + --| = 5
\ 3 /
$$2 \cos{\left(x + \frac{\pi}{3} \right)} = 5$$
Detail solution
Given the equation
$$2 \cos{\left(x + \frac{\pi}{3} \right)} = 5$$
- this is the simplest trigonometric equation
The equation is transformed to
$$\cos{\left(x + \frac{\pi}{3} \right)} = \frac{5}{2}$$
As right part of the equation
modulo =
but cos
can no be more than 1 or less than -1
so the solution of the equation d'not exist.
$$2 \cos{\left(x + \frac{\pi}{3} \right)} = 5$$
- this is the simplest trigonometric equation
Divide both parts of the equation by 2
The equation is transformed to
$$\cos{\left(x + \frac{\pi}{3} \right)} = \frac{5}{2}$$
As right part of the equation
modulo =
True
but cos
can no be more than 1 or less than -1
so the solution of the equation d'not exist.
Rapid solution
[src]
pi
x1 = - -- + I*im(acos(5/2)) + re(acos(5/2))
3
$$x_{1} = - \frac{\pi}{3} + \operatorname{re}{\left(\operatorname{acos}{\left(\frac{5}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{5}{2} \right)}\right)}$$
5*pi
x2 = ---- - I*im(acos(5/2))
3
$$x_{2} = \frac{5 \pi}{3} - i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{5}{2} \right)}\right)}$$
x2 = 5*pi/3 - i*im(acos(5/2))
Sum and product of roots
[src]
sum
pi 5*pi - -- + I*im(acos(5/2)) + re(acos(5/2)) + ---- - I*im(acos(5/2)) 3 3
$$\left(\frac{5 \pi}{3} - i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{5}{2} \right)}\right)}\right) + \left(- \frac{\pi}{3} + \operatorname{re}{\left(\operatorname{acos}{\left(\frac{5}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{5}{2} \right)}\right)}\right)$$
=
4*pi ---- + re(acos(5/2)) 3
$$\operatorname{re}{\left(\operatorname{acos}{\left(\frac{5}{2} \right)}\right)} + \frac{4 \pi}{3}$$
product
/ pi \ /5*pi \ |- -- + I*im(acos(5/2)) + re(acos(5/2))|*|---- - I*im(acos(5/2))| \ 3 / \ 3 /
$$\left(\frac{5 \pi}{3} - i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{5}{2} \right)}\right)}\right) \left(- \frac{\pi}{3} + \operatorname{re}{\left(\operatorname{acos}{\left(\frac{5}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{5}{2} \right)}\right)}\right)$$
=
(5*pi - 3*I*im(acos(5/2)))*(-pi + 3*re(acos(5/2)) + 3*I*im(acos(5/2)))
----------------------------------------------------------------------
9
$$\frac{\left(5 \pi - 3 i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{5}{2} \right)}\right)}\right) \left(- \pi + 3 \operatorname{re}{\left(\operatorname{acos}{\left(\frac{5}{2} \right)}\right)} + 3 i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{5}{2} \right)}\right)}\right)}{9}$$
(5*pi - 3*i*im(acos(5/2)))*(-pi + 3*re(acos(5/2)) + 3*i*im(acos(5/2)))/9
Numerical answer
[src]
x1 = -1.0471975511966 + 1.56679923697241*i
x2 = 5.23598775598299 - 1.56679923697241*i
x2 = 5.23598775598299 - 1.56679923697241*i