Mister Exam
Other calculators
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How to use it?
- Equation:
-
Equation 49*x^3+14*x^2+x=0
-
Equation 4^x=8
-
Equation x^2-x=12
-
Equation x-5(x+3)=5
- Express {x} in terms of y where:
- -7*x-13*y=-12
- -3*x+19*y=16
- -4*x+4*y=-9
- -19*x-2*y=-6
- Derivative of:
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cos6x
- Integral of d{x}:
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cos6x
-
Identical expressions
- cos6x=-sqrt(three / two)
- co sinus of e of 6x equally minus square root of (3 divide by 2)
- co sinus of e of 6x equally minus square root of (three divide by two)
- cos6x=-√(3/2)
- cos6x=-sqrt3/2
- cos6x=-sqrt(3 divide by 2)
-
Similar expressions
- sin(6*x)+(3^(1/2))*(cos(6*x))+2*(cos(8*x))=0
- 6*sin(5x)+5*cos(6x)=-11
- (sin(2*x)-(sqrt(3))/2)^2+(sin(x)-1/2)^2=0
- cos(2*x)+cos(8*x)-cos(6*x)=1
- sin(x)-sqrt(3)/2=0
- cos(2*x-pi/3)=(-sqrt(3))/2
- cos(6*x)-cos(8*x)=1-cos(2*x)
- cos(p/3+x)=(-sqrt(3))/2
- cos6x=+sqrt(3/2)
cos6x=-sqrt(3/2) equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation
$$\cos{\left(6 x \right)} = - \sqrt{\frac{3}{2}}$$
- this is the simplest trigonometric equation
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.
$$\cos{\left(6 x \right)} = - \sqrt{\frac{3}{2}}$$
- this is the simplest trigonometric equation
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]
/ / ___ \\ / / ___ \\
| |-\/ 6 || | |-\/ 6 ||
re|acos|-------|| I*im|acos|-------||
\ \ 2 // pi \ \ 2 //
x1 = - ----------------- + -- - -------------------
6 3 6
$$x_{1} = - \frac{\operatorname{re}{\left(\operatorname{acos}{\left(- \frac{\sqrt{6}}{2} \right)}\right)}}{6} + \frac{\pi}{3} - \frac{i \operatorname{im}{\left(\operatorname{acos}{\left(- \frac{\sqrt{6}}{2} \right)}\right)}}{6}$$
/ / ___ \\ / / ___ \\
| |-\/ 6 || | |-\/ 6 ||
re|acos|-------|| I*im|acos|-------||
\ \ 2 // \ \ 2 //
x2 = ----------------- + -------------------
6 6
$$x_{2} = \frac{\operatorname{re}{\left(\operatorname{acos}{\left(- \frac{\sqrt{6}}{2} \right)}\right)}}{6} + \frac{i \operatorname{im}{\left(\operatorname{acos}{\left(- \frac{\sqrt{6}}{2} \right)}\right)}}{6}$$
x2 = re(acos(-sqrt(6)/2))/6 + i*im(acos(-sqrt(6)/2))/6
Sum and product of roots
[src]
sum
/ / ___ \\ / / ___ \\ / / ___ \\ / / ___ \\
| |-\/ 6 || | |-\/ 6 || | |-\/ 6 || | |-\/ 6 ||
re|acos|-------|| I*im|acos|-------|| re|acos|-------|| I*im|acos|-------||
\ \ 2 // pi \ \ 2 // \ \ 2 // \ \ 2 //
- ----------------- + -- - ------------------- + ----------------- + -------------------
6 3 6 6 6
$$\left(\frac{\operatorname{re}{\left(\operatorname{acos}{\left(- \frac{\sqrt{6}}{2} \right)}\right)}}{6} + \frac{i \operatorname{im}{\left(\operatorname{acos}{\left(- \frac{\sqrt{6}}{2} \right)}\right)}}{6}\right) + \left(- \frac{\operatorname{re}{\left(\operatorname{acos}{\left(- \frac{\sqrt{6}}{2} \right)}\right)}}{6} + \frac{\pi}{3} - \frac{i \operatorname{im}{\left(\operatorname{acos}{\left(- \frac{\sqrt{6}}{2} \right)}\right)}}{6}\right)$$
=
pi -- 3
$$\frac{\pi}{3}$$
product
/ / / ___ \\ / / ___ \\\ / / / ___ \\ / / ___ \\\ | | |-\/ 6 || | |-\/ 6 ||| | | |-\/ 6 || | |-\/ 6 ||| | re|acos|-------|| I*im|acos|-------||| |re|acos|-------|| I*im|acos|-------||| | \ \ 2 // pi \ \ 2 //| | \ \ 2 // \ \ 2 //| |- ----------------- + -- - -------------------|*|----------------- + -------------------| \ 6 3 6 / \ 6 6 /
$$\left(\frac{\operatorname{re}{\left(\operatorname{acos}{\left(- \frac{\sqrt{6}}{2} \right)}\right)}}{6} + \frac{i \operatorname{im}{\left(\operatorname{acos}{\left(- \frac{\sqrt{6}}{2} \right)}\right)}}{6}\right) \left(- \frac{\operatorname{re}{\left(\operatorname{acos}{\left(- \frac{\sqrt{6}}{2} \right)}\right)}}{6} + \frac{\pi}{3} - \frac{i \operatorname{im}{\left(\operatorname{acos}{\left(- \frac{\sqrt{6}}{2} \right)}\right)}}{6}\right)$$
=
/ / / ___ \\ / / ___ \\\ / / / ___ \\ / / ___ \\\
| | |-\/ 6 || | |-\/ 6 ||| | | |-\/ 6 || | |-\/ 6 |||
-|I*im|acos|-------|| + re|acos|-------|||*|-2*pi + I*im|acos|-------|| + re|acos|-------|||
\ \ \ 2 // \ \ 2 /// \ \ \ 2 // \ \ 2 ///
---------------------------------------------------------------------------------------------
36
$$- \frac{\left(\operatorname{re}{\left(\operatorname{acos}{\left(- \frac{\sqrt{6}}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(- \frac{\sqrt{6}}{2} \right)}\right)}\right) \left(- 2 \pi + \operatorname{re}{\left(\operatorname{acos}{\left(- \frac{\sqrt{6}}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(- \frac{\sqrt{6}}{2} \right)}\right)}\right)}{36}$$
-(i*im(acos(-sqrt(6)/2)) + re(acos(-sqrt(6)/2)))*(-2*pi + i*im(acos(-sqrt(6)/2)) + re(acos(-sqrt(6)/2)))/36
Numerical answer
[src]
x1 = 0.523598775598299 + 0.109746491410401*i
x2 = 0.523598775598299 - 0.109746491410401*i
x2 = 0.523598775598299 - 0.109746491410401*i